Theorem sorpssin 7584

Description: A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
sorpssin	⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∩ 𝐶) ∈ 𝐴)

Proof of Theorem sorpssin

Step	Hyp	Ref	Expression
1		simprl 768	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
2		df-ss 3904	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∩ 𝐶) = 𝐵)
3		eleq1 2826	. . . 4 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4	2, 3	sylbi 216	. . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
5	1, 4	syl5ibrcom 246	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 → (𝐵 ∩ 𝐶) ∈ 𝐴))
6		simprr 770	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
7		sseqin2 4149	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶)
8		eleq1 2826	. . . 4 ⊢ ((𝐵 ∩ 𝐶) = 𝐶 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
9	7, 8	sylbi 216	. . 3 ⊢ (𝐶 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
10	6, 9	syl5ibrcom 246	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶 ⊆ 𝐵 → (𝐵 ∩ 𝐶) ∈ 𝐴))
11		sorpssi 7582	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
12	5, 10, 11	mpjaod 857	1 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∩ 𝐶) ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∩ cin 3886   ⊆ wss 3887   Or wor 5502   [⊊] crpss 7575

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-so 5504  df-xp 5595  df-rel 5596  df-rpss 7576

This theorem is referenced by: (None)