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Theorem sorpssi 6908
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 5028 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 [] 𝐶𝐵 = 𝐶𝐶 [] 𝐵))
2 elex 3202 . . . . . 6 (𝐶𝐴𝐶 ∈ V)
32ad2antll 764 . . . . 5 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶 ∈ V)
4 brrpssg 6904 . . . . 5 (𝐶 ∈ V → (𝐵 [] 𝐶𝐵𝐶))
53, 4syl 17 . . . 4 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 [] 𝐶𝐵𝐶))
6 biidd 252 . . . 4 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 = 𝐶𝐵 = 𝐶))
7 elex 3202 . . . . . 6 (𝐵𝐴𝐵 ∈ V)
87ad2antrl 763 . . . . 5 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵 ∈ V)
9 brrpssg 6904 . . . . 5 (𝐵 ∈ V → (𝐶 [] 𝐵𝐶𝐵))
108, 9syl 17 . . . 4 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶 [] 𝐵𝐶𝐵))
115, 6, 103orbi123d 1395 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 [] 𝐶𝐵 = 𝐶𝐶 [] 𝐵) ↔ (𝐵𝐶𝐵 = 𝐶𝐶𝐵)))
121, 11mpbid 222 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐵 = 𝐶𝐶𝐵))
13 sspsstri 3693 . 2 ((𝐵𝐶𝐶𝐵) ↔ (𝐵𝐶𝐵 = 𝐶𝐶𝐵))
1412, 13sylibr 224 1 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3o 1035   = wceq 1480  wcel 1987  Vcvv 3190  wss 3560  wpss 3561   class class class wbr 4623   Or wor 5004   [] crpss 6901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-so 5006  df-xp 5090  df-rel 5091  df-rpss 6902
This theorem is referenced by:  sorpssun  6909  sorpssin  6910  sorpssuni  6911  sorpssint  6912  sorpsscmpl  6913  enfin2i  9103  fin1a2lem9  9190  fin1a2lem10  9191  fin1a2lem11  9192  fin1a2lem13  9194
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