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Theorem sorpssun 6904
Description: A chain of sets is closed under binary union. (Contributed by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
sorpssun (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)

Proof of Theorem sorpssun
StepHypRef Expression
1 simprr 795 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶𝐴)
2 ssequn1 3766 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
3 eleq1 2686 . . . 4 ((𝐵𝐶) = 𝐶 → ((𝐵𝐶) ∈ 𝐴𝐶𝐴))
42, 3sylbi 207 . . 3 (𝐵𝐶 → ((𝐵𝐶) ∈ 𝐴𝐶𝐴))
51, 4syl5ibrcom 237 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶 → (𝐵𝐶) ∈ 𝐴))
6 simprl 793 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵𝐴)
7 ssequn2 3769 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
8 eleq1 2686 . . . 4 ((𝐵𝐶) = 𝐵 → ((𝐵𝐶) ∈ 𝐴𝐵𝐴))
97, 8sylbi 207 . . 3 (𝐶𝐵 → ((𝐵𝐶) ∈ 𝐴𝐵𝐴))
106, 9syl5ibrcom 237 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶𝐵 → (𝐵𝐶) ∈ 𝐴))
11 sorpssi 6903 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))
125, 10, 11mpjaod 396 1 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cun 3557  wss 3559   Or wor 4999   [] crpss 6896
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 4746  ax-nul 4754  ax-pr 4872
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 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-so 5001  df-xp 5085  df-rel 5086  df-rpss 6897
This theorem is referenced by:  finsschain  8225  lbsextlem2  19091  lbsextlem3  19092  filssufilg  21638
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