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Theorem sorpssi 7764
Description: Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
sorpssi (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))

Proof of Theorem sorpssi
StepHypRef Expression
1 solin 5634 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 [] 𝐶𝐵 = 𝐶𝐶 [] 𝐵))
2 elex 3509 . . . . . 6 (𝐶𝐴𝐶 ∈ V)
32ad2antll 728 . . . . 5 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶 ∈ V)
4 brrpssg 7760 . . . . 5 (𝐶 ∈ V → (𝐵 [] 𝐶𝐵𝐶))
53, 4syl 17 . . . 4 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 [] 𝐶𝐵𝐶))
6 biidd 262 . . . 4 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶𝐵 = 𝐶))
7 elex 3509 . . . . . 6 (𝐵𝐴𝐵 ∈ V)
87ad2antrl 727 . . . . 5 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵 ∈ V)
9 brrpssg 7760 . . . . 5 (𝐵 ∈ V → (𝐶 [] 𝐵𝐶𝐵))
108, 9syl 17 . . . 4 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶 [] 𝐵𝐶𝐵))
115, 6, 103orbi123d 1435 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 [] 𝐶𝐵 = 𝐶𝐶 [] 𝐵) ↔ (𝐵𝐶𝐵 = 𝐶𝐶𝐵)))
121, 11mpbid 232 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐵 = 𝐶𝐶𝐵))
13 sspsstri 4128 . 2 ((𝐵𝐶𝐶𝐵) ↔ (𝐵𝐶𝐵 = 𝐶𝐶𝐵))
1412, 13sylibr 234 1 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 846  w3o 1086   = wceq 1537  wcel 2108  Vcvv 3488  wss 3976  wpss 3977   class class class wbr 5166   Or wor 5606   [] crpss 7757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-so 5608  df-xp 5706  df-rel 5707  df-rpss 7758
This theorem is referenced by:  sorpssun  7765  sorpssin  7766  sorpssuni  7767  sorpssint  7768  sorpsscmpl  7769  enfin2i  10390  fin1a2lem9  10477  fin1a2lem10  10478  fin1a2lem11  10479  fin1a2lem13  10481  ssdifidllem  33449  ssmxidllem  33466
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