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Theorem sorpssin 7725
Description: A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
sorpssin (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)

Proof of Theorem sorpssin
StepHypRef Expression
1 simprl 768 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵𝐴)
2 df-ss 3965 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐵)
3 eleq1 2820 . . . 4 ((𝐵𝐶) = 𝐵 → ((𝐵𝐶) ∈ 𝐴𝐵𝐴))
42, 3sylbi 216 . . 3 (𝐵𝐶 → ((𝐵𝐶) ∈ 𝐴𝐵𝐴))
51, 4syl5ibrcom 246 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶 → (𝐵𝐶) ∈ 𝐴))
6 simprr 770 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶𝐴)
7 sseqin2 4215 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐶)
8 eleq1 2820 . . . 4 ((𝐵𝐶) = 𝐶 → ((𝐵𝐶) ∈ 𝐴𝐶𝐴))
97, 8sylbi 216 . . 3 (𝐶𝐵 → ((𝐵𝐶) ∈ 𝐴𝐶𝐴))
106, 9syl5ibrcom 246 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶𝐵 → (𝐵𝐶) ∈ 𝐴))
11 sorpssi 7723 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))
125, 10, 11mpjaod 857 1 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  cin 3947  wss 3948   Or wor 5587   [] crpss 7716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-so 5589  df-xp 5682  df-rel 5683  df-rpss 7717
This theorem is referenced by: (None)
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