Theorem sorpssun 7669

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 4135	. . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶)
3		eleq1 2821	. . . 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 4138	. . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵)
8		eleq1 2821	. . . 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 7668	. 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 1541   ∈ wcel 2113   ∪ cun 3896   ⊆ wss 3898   Or wor 5526   [⊊] crpss 7661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-so 5528  df-xp 5625  df-rel 5626  df-rpss 7662

This theorem is referenced by:  finsschain  9250  lbsextlem2  21098  lbsextlem3  21099  filssufilg  23827