Theorem sorpssin 7437

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		df-ss 3898	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∩ 𝐶) = 𝐵)
3		eleq1 2877	. . . 4 ⊢ ((𝐵 ∩ 𝐶) = 𝐵 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
4	2, 3	sylbi 220	. . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
5	1, 4	syl5ibrcom 250	. 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 4142	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐶) = 𝐶)
8		eleq1 2877	. . . 4 ⊢ ((𝐵 ∩ 𝐶) = 𝐶 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
9	7, 8	sylbi 220	. . 3 ⊢ (𝐶 ⊆ 𝐵 → ((𝐵 ∩ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
10	6, 9	syl5ibrcom 250	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶 ⊆ 𝐵 → (𝐵 ∩ 𝐶) ∈ 𝐴))
11		sorpssi 7435	. 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∩ cin 3880   ⊆ wss 3881   Or wor 5437   [⊊] crpss 7428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-so 5439  df-xp 5525  df-rel 5526  df-rpss 7429

This theorem is referenced by: (None)