Theorem sorpssin 7707

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 770	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
2		dfss2 3932	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∩ 𝐶) = 𝐵)
3		eleq1 2816	. . . 4 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4	2, 3	sylbi 217	. . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
5	1, 4	syl5ibrcom 247	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 → (𝐵 ∩ 𝐶) ∈ 𝐴))
6		simprr 772	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
7		sseqin2 4186	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶)
8		eleq1 2816	. . . 4 ⊢ ((𝐵 ∩ 𝐶) = 𝐶 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
9	7, 8	sylbi 217	. . 3 ⊢ (𝐶 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
10	6, 9	syl5ibrcom 247	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶 ⊆ 𝐵 → (𝐵 ∩ 𝐶) ∈ 𝐴))
11		sorpssi 7705	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
12	5, 10, 11	mpjaod 860	1 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∩ 𝐶) ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3913   ⊆ wss 3914   Or wor 5545   [⊊] crpss 7698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-so 5547  df-xp 5644  df-rel 5645  df-rpss 7699

This theorem is referenced by: (None)