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Theorem sstrALT2 41538
Description: Virtual deduction proof of sstr 3926, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 41537 using the command file translate_without_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 3904 . 2 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
2 id 22 . . . . . 6 ((𝐴𝐵𝐵𝐶) → (𝐴𝐵𝐵𝐶))
3 simpr 488 . . . . . 6 ((𝐴𝐵𝐵𝐶) → 𝐵𝐶)
42, 3syl 17 . . . . 5 ((𝐴𝐵𝐵𝐶) → 𝐵𝐶)
5 simpl 486 . . . . . . 7 ((𝐴𝐵𝐵𝐶) → 𝐴𝐵)
62, 5syl 17 . . . . . 6 ((𝐴𝐵𝐵𝐶) → 𝐴𝐵)
7 idd 24 . . . . . 6 ((𝐴𝐵𝐵𝐶) → (𝑥𝐴𝑥𝐴))
8 ssel2 3913 . . . . . 6 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
96, 7, 8syl6an 683 . . . . 5 ((𝐴𝐵𝐵𝐶) → (𝑥𝐴𝑥𝐵))
10 ssel2 3913 . . . . 5 ((𝐵𝐶𝑥𝐵) → 𝑥𝐶)
114, 9, 10syl6an 683 . . . 4 ((𝐴𝐵𝐵𝐶) → (𝑥𝐴𝑥𝐶))
1211idiALT 41180 . . 3 ((𝐴𝐵𝐵𝐶) → (𝑥𝐴𝑥𝐶))
1312alrimiv 1928 . 2 ((𝐴𝐵𝐵𝐶) → ∀𝑥(𝑥𝐴𝑥𝐶))
14 biimpr 223 . 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 209  wa 399  wal 1536  wcel 2112  wss 3884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator