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Theorem sorpssun 7209
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 789 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶𝐴)
2 ssequn1 4012 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
3 eleq1 2894 . . . 4 ((𝐵𝐶) = 𝐶 → ((𝐵𝐶) ∈ 𝐴𝐶𝐴))
42, 3sylbi 209 . . 3 (𝐵𝐶 → ((𝐵𝐶) ∈ 𝐴𝐶𝐴))
51, 4syl5ibrcom 239 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶 → (𝐵𝐶) ∈ 𝐴))
6 simprl 787 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵𝐴)
7 ssequn2 4015 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
8 eleq1 2894 . . . 4 ((𝐵𝐶) = 𝐵 → ((𝐵𝐶) ∈ 𝐴𝐵𝐴))
97, 8sylbi 209 . . 3 (𝐶𝐵 → ((𝐵𝐶) ∈ 𝐴𝐵𝐴))
106, 9syl5ibrcom 239 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶𝐵 → (𝐵𝐶) ∈ 𝐴))
11 sorpssi 7208 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))
125, 10, 11mpjaod 891 1 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  cun 3796  wss 3798   Or wor 5264   [] crpss 7201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-so 5266  df-xp 5352  df-rel 5353  df-rpss 7202
This theorem is referenced by:  finsschain  8548  lbsextlem2  19527  lbsextlem3  19528  filssufilg  22092
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