Theorem sorpssun 7209

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

Assertion

Ref	Expression
sorpssun	⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∪ 𝐶) ∈ 𝐴)

Proof of Theorem sorpssun

Step	Hyp	Ref	Expression
1		simprr 789	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
2		ssequn1 4012	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶)
3		eleq1 2894	. . . 4 ⊢ ((𝐵 ∪ 𝐶) = 𝐶 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
4	2, 3	sylbi 209	. . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
5	1, 4	syl5ibrcom 239	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 → (𝐵 ∪ 𝐶) ∈ 𝐴))
6		simprl 787	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
7		ssequn2 4015	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵)
8		eleq1 2894	. . . 4 ⊢ ((𝐵 ∪ 𝐶) = 𝐵 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
9	7, 8	sylbi 209	. . 3 ⊢ (𝐶 ⊆ 𝐵 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
10	6, 9	syl5ibrcom 239	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶 ⊆ 𝐵 → (𝐵 ∪ 𝐶) ∈ 𝐴))
11		sorpssi 7208	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
12	5, 10, 11	mpjaod 891	1 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∪ 𝐶) ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164   ∪ cun 3796   ⊆ wss 3798   Or wor 5264   [⊊] crpss 7201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-so 5266  df-xp 5352  df-rel 5353  df-rpss 7202

This theorem is referenced by:  finsschain  8548  lbsextlem2  19527  lbsextlem3  19528  filssufilg  22092