Theorem sorpssun 7717

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 784	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ 𝐴)
2		ssequn1 4141	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶)
3		eleq1 2853	. . . 4 ⊢ ((𝐵 ∪ 𝐶) = 𝐶 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
4	2, 3	sylbi 220	. . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
5	1, 4	syl5ibrcom 250	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 → (𝐵 ∪ 𝐶) ∈ 𝐴))
6		simprl 782	. . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴)
7		ssequn2 4144	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵)
8		eleq1 2853	. . . 4 ⊢ ((𝐵 ∪ 𝐶) = 𝐵 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
9	7, 8	sylbi 220	. . 3 ⊢ (𝐶 ⊆ 𝐵 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
10	6, 9	syl5ibrcom 250	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶 ⊆ 𝐵 → (𝐵 ∪ 𝐶) ∈ 𝐴))
11		sorpssi 7716	. 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵))
12	5, 10, 11	mpjaod 873	1 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∪ 𝐶) ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ∪ cun 3905   ⊆ wss 3907   Or wor 5558   [⊊] crpss 7709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-so 5560  df-xp 5657  df-rel 5658  df-rpss 7710

This theorem is referenced by:  finsschain  9304  lbsextlem2  21249  lbsextlem3  21250  filssufilg  24025