Theorem sorpssun 7751

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 772	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
2		ssequn1 4185	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶)
3		eleq1 2828	. . . 4 ⊢ ((𝐵 ∪ 𝐶) = 𝐶 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
4	2, 3	sylbi 217	. . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
5	1, 4	syl5ibrcom 247	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 → (𝐵 ∪ 𝐶) ∈ 𝐴))
6		simprl 770	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
7		ssequn2 4188	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵)
8		eleq1 2828	. . . 4 ⊢ ((𝐵 ∪ 𝐶) = 𝐵 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
9	7, 8	sylbi 217	. . 3 ⊢ (𝐶 ⊆ 𝐵 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
10	6, 9	syl5ibrcom 247	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶 ⊆ 𝐵 → (𝐵 ∪ 𝐶) ∈ 𝐴))
11		sorpssi 7750	. 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 1539   ∈ wcel 2107   ∪ cun 3948   ⊆ wss 3950   Or wor 5590   [⊊] crpss 7743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-so 5592  df-xp 5690  df-rel 5691  df-rpss 7744

This theorem is referenced by:  finsschain  9400  lbsextlem2  21162  lbsextlem3  21163  filssufilg  23920