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Theorem sorpssin 7669
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 770 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐵𝐴)
2 df-ss 3928 . . . 4 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐵)
3 eleq1 2826 . . . 4 ((𝐵𝐶) = 𝐵 → ((𝐵𝐶) ∈ 𝐴𝐵𝐴))
42, 3sylbi 216 . . 3 (𝐵𝐶 → ((𝐵𝐶) ∈ 𝐴𝐵𝐴))
51, 4syl5ibrcom 247 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶 → (𝐵𝐶) ∈ 𝐴))
6 simprr 772 . . 3 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → 𝐶𝐴)
7 sseqin2 4176 . . . 4 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐶)
8 eleq1 2826 . . . 4 ((𝐵𝐶) = 𝐶 → ((𝐵𝐶) ∈ 𝐴𝐶𝐴))
97, 8sylbi 216 . . 3 (𝐶𝐵 → ((𝐵𝐶) ∈ 𝐴𝐶𝐴))
106, 9syl5ibrcom 247 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐶𝐵 → (𝐵𝐶) ∈ 𝐴))
11 sorpssi 7667 . 2 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))
125, 10, 11mpjaod 859 1 (( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  cin 3910  wss 3911   Or wor 5545   [] crpss 7660
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 5257  ax-nul 5264  ax-pr 5385
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 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-so 5547  df-xp 5640  df-rel 5641  df-rpss 7661
This theorem is referenced by: (None)
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