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Theorem sorpssi 7671
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 5575 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵 [] 𝐶𝐵 = 𝐶𝐶 [] 𝐵))
2 elex 3466 . . . . . 6 (𝐶𝐴𝐶 ∈ V)
32ad2antll 728 . . . . 5 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶 ∈ V)
4 brrpssg 7667 . . . . 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 3466 . . . . . 6 (𝐵𝐴𝐵 ∈ V)
87ad2antrl 727 . . . . 5 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵 ∈ V)
9 brrpssg 7667 . . . . 5 (𝐵 ∈ V → (𝐶 [] 𝐵𝐶𝐵))
108, 9syl 17 . . . 4 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶 [] 𝐵𝐶𝐵))
115, 6, 103orbi123d 1436 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → ((𝐵 [] 𝐶𝐵 = 𝐶𝐶 [] 𝐵) ↔ (𝐵𝐶𝐵 = 𝐶𝐶𝐵)))
121, 11mpbid 231 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐵 = 𝐶𝐶𝐵))
13 sspsstri 4067 . 2 ((𝐵𝐶𝐶𝐵) ↔ (𝐵𝐶𝐵 = 𝐶𝐶𝐵))
1412, 13sylibr 233 1 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  w3o 1087   = wceq 1542  wcel 2107  Vcvv 3448  wss 3915  wpss 3916   class class class wbr 5110   Or wor 5549   [] crpss 7664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-so 5551  df-xp 5644  df-rel 5645  df-rpss 7665
This theorem is referenced by:  sorpssun  7672  sorpssin  7673  sorpssuni  7674  sorpssint  7675  sorpsscmpl  7676  enfin2i  10264  fin1a2lem9  10351  fin1a2lem10  10352  fin1a2lem11  10353  fin1a2lem13  10355  ssmxidllem  32278
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