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Theorem sorpssun 7736
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 771 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶𝐴)
2 ssequn1 4178 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
3 eleq1 2813 . . . 4 ((𝐵𝐶) = 𝐶 → ((𝐵𝐶) ∈ 𝐴𝐶𝐴))
42, 3sylbi 216 . . 3 (𝐵𝐶 → ((𝐵𝐶) ∈ 𝐴𝐶𝐴))
51, 4syl5ibrcom 246 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶 → (𝐵𝐶) ∈ 𝐴))
6 simprl 769 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵𝐴)
7 ssequn2 4181 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
8 eleq1 2813 . . . 4 ((𝐵𝐶) = 𝐵 → ((𝐵𝐶) ∈ 𝐴𝐵𝐴))
97, 8sylbi 216 . . 3 (𝐶𝐵 → ((𝐵𝐶) ∈ 𝐴𝐵𝐴))
106, 9syl5ibrcom 246 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶𝐵 → (𝐵𝐶) ∈ 𝐴))
11 sorpssi 7735 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))
125, 10, 11mpjaod 858 1 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  cun 3942  wss 3944   Or wor 5589   [] crpss 7728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-so 5591  df-xp 5684  df-rel 5685  df-rpss 7729
This theorem is referenced by:  finsschain  9385  lbsextlem2  21059  lbsextlem3  21060  filssufilg  23859
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