Can you remind me what the minimal positive calculus is? On Friday, October 29, 2021 at 11:58:02 AM UTC-4 Benoit wrote:
> The hypothesis of bj-df-cleq corresponds to setvar substitutions into A >> and B. Do we know that other substitutions into A and B can't generate FOL= >> statements that are stronger than the hypothesis? Perhaps we are safe since >> we may need df-cleq itself to eliminate the resulting class notation, but I >> haven't thought it through completely to be sure there aren't tricks that >> can be done with the other two class definitions. What do you think? > > > If I understand your question correctly, then the following result might > answer your concerns: > Theorem: Let S be any set of schemes. Assume that we can prove a scheme > Phi which contains no class terms or class variables from S \cup \{df-clab, > bj-df-cleq, bj-df-clel\}. Then, we can prove it from S \cup {minimal > positive calculus}. > Proof: Consider a proof of Phi from S \cup \{df-clab, bj-df-cleq, > bj-df-clel\}. Using \{df-clab, bj-df-cleq, bj-df-clel, minimal positive > calculus\}, eliminate class abstractions from the proof. There may also be > dummy class variables; replace them with fresh dummy setvar variables. > Now, the proof contains no class abstraction so does not use df-clab, and > for the lines using bj-df-cleq or bj-df-clel, they can be "removed" since > the conclusion is an alpha-renaming of the hypthesis (more precisely, you > prove their conclusion directly using the corresponding alpha-renaming of > the proof of its hypothesis in the original proof). QED > > BenoƮt > -- You received this message because you are subscribed to the Google Groups "Metamath" group. To unsubscribe from this group and stop receiving emails from it, send an email to metamath+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/metamath/d6e3628d-8f02-4b72-9c1d-24809f11e78bn%40googlegroups.com.