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Theorem sorpssi 7749
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 5619 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 [] 𝐶𝐵 = 𝐶𝐶 [] 𝐵))
2 elex 3501 . . . . . 6 (𝐶𝐴𝐶 ∈ V)
32ad2antll 729 . . . . 5 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶 ∈ V)
4 brrpssg 7745 . . . . 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 3501 . . . . . 6 (𝐵𝐴𝐵 ∈ V)
87ad2antrl 728 . . . . 5 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵 ∈ V)
9 brrpssg 7745 . . . . 5 (𝐵 ∈ V → (𝐶 [] 𝐵𝐶𝐵))
108, 9syl 17 . . . 4 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶 [] 𝐵𝐶𝐵))
115, 6, 103orbi123d 1437 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 [] 𝐶𝐵 = 𝐶𝐶 [] 𝐵) ↔ (𝐵𝐶𝐵 = 𝐶𝐶𝐵)))
121, 11mpbid 232 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐵 = 𝐶𝐶𝐵))
13 sspsstri 4105 . 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 848  w3o 1086   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  wpss 3952   class class class wbr 5143   Or wor 5591   [] crpss 7742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-so 5593  df-xp 5691  df-rel 5692  df-rpss 7743
This theorem is referenced by:  sorpssun  7750  sorpssin  7751  sorpssuni  7752  sorpssint  7753  sorpsscmpl  7754  enfin2i  10361  fin1a2lem9  10448  fin1a2lem10  10449  fin1a2lem11  10450  fin1a2lem13  10452  ssdifidllem  33484  ssmxidllem  33501
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