Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sstrALT2 Structured version   Visualization version   GIF version

Theorem sstrALT2 39565
Description: Virtual deduction proof of sstr 3813, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 39564 using the command file translatewithout_overwriting.cmd . It was not minimized because the automated minimization excluding duplicates generates a minimized proof which, although not directly containing any duplicates, indirectly contains a duplicate. That is, the trace back of the minimized proof contains a duplicate. This is undesirable because some step(s) of the minimized proof use the proven theorem. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sstrALT2 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Proof of Theorem sstrALT2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3793 . 2 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
2 id 22 . . . . . 6 ((𝐴𝐵𝐵𝐶) → (𝐴𝐵𝐵𝐶))
3 simpr 473 . . . . . 6 ((𝐴𝐵𝐵𝐶) → 𝐵𝐶)
42, 3syl 17 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐵𝐶)
5 simpl 470 . . . . . . 7 ((𝐴𝐵𝐵𝐶) → 𝐴𝐵)
62, 5syl 17 . . . . . 6 ((𝐴𝐵𝐵𝐶) → 𝐴𝐵)
7 idd 24 . . . . . 6 ((𝐴𝐵𝐵𝐶) → (𝑥𝐴𝑥𝐴))
8 ssel2 3800 . . . . . 6 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
96, 7, 8syl6an 666 . . . . 5 ((𝐴𝐵𝐵𝐶) → (𝑥𝐴𝑥𝐵))
10 ssel2 3800 . . . . 5 ((𝐵𝐶𝑥𝐵) → 𝑥𝐶)
114, 9, 10syl6an 666 . . . 4 ((𝐴𝐵𝐵𝐶) → (𝑥𝐴𝑥𝐶))
1211idiALT 39182 . . 3 ((𝐴𝐵𝐵𝐶) → (𝑥𝐴𝑥𝐶))
1312alrimiv 2018 . 2 ((𝐴𝐵𝐵𝐶) → ∀𝑥(𝑥𝐴𝑥𝐶))
14 biimpr 211 . 2 ((𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶)) → (∀𝑥(𝑥𝐴𝑥𝐶) → 𝐴𝐶))
151, 13, 14mpsyl 68 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635  wcel 2157  wss 3776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-in 3783  df-ss 3790
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator