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Mirrors > Home > MPE Home > Th. List > sorpssun | Structured version Visualization version GIF version |
Description: A chain of sets is closed under binary union. (Contributed by Mario Carneiro, 16-May-2015.) |
Ref | Expression |
---|---|
sorpssun | ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∪ 𝐶) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 771 | . . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐶 ∈ 𝐴) | |
2 | ssequn1 4178 | . . . 4 ⊢ (𝐵 ⊆ 𝐶 ↔ (𝐵 ∪ 𝐶) = 𝐶) | |
3 | eleq1 2813 | . . . 4 ⊢ ((𝐵 ∪ 𝐶) = 𝐶 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
4 | 2, 3 | sylbi 216 | . . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) |
5 | 1, 4 | syl5ibrcom 246 | . 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 → (𝐵 ∪ 𝐶) ∈ 𝐴)) |
6 | simprl 769 | . . 3 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → 𝐵 ∈ 𝐴) | |
7 | ssequn2 4181 | . . . 4 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∪ 𝐶) = 𝐵) | |
8 | eleq1 2813 | . . . 4 ⊢ ((𝐵 ∪ 𝐶) = 𝐵 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
9 | 7, 8 | sylbi 216 | . . 3 ⊢ (𝐶 ⊆ 𝐵 → ((𝐵 ∪ 𝐶) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) |
10 | 6, 9 | syl5ibrcom 246 | . 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶 ⊆ 𝐵 → (𝐵 ∪ 𝐶) ∈ 𝐴)) |
11 | sorpssi 7735 | . 2 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) | |
12 | 5, 10, 11 | mpjaod 858 | 1 ⊢ (( [⊊] Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ∪ 𝐶) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∪ cun 3942 ⊆ wss 3944 Or wor 5589 [⊊] crpss 7728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-so 5591 df-xp 5684 df-rel 5685 df-rpss 7729 |
This theorem is referenced by: finsschain 9385 lbsextlem2 21059 lbsextlem3 21060 filssufilg 23859 |
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