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Mirrors > Home > MPE Home > Th. List > compsscnvlem | Structured version Visualization version GIF version |
Description: Lemma for compsscnv 10396. (Contributed by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
compsscnvlem | ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 = (𝐴 ∖ 𝑥)) | |
2 | difss 4128 | . . . 4 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴 | |
3 | 1, 2 | eqsstrdi 4031 | . . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 ⊆ 𝐴) |
4 | velpw 4609 | . . 3 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴) | |
5 | 3, 4 | sylibr 233 | . 2 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 ∈ 𝒫 𝐴) |
6 | 1 | difeq2d 4118 | . . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝐴 ∖ 𝑦) = (𝐴 ∖ (𝐴 ∖ 𝑥))) |
7 | elpwi 4611 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
8 | 7 | adantr 479 | . . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑥 ⊆ 𝐴) |
9 | dfss4 4257 | . . . 4 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥) | |
10 | 8, 9 | sylib 217 | . . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥) |
11 | 6, 10 | eqtr2d 2766 | . 2 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑥 = (𝐴 ∖ 𝑦)) |
12 | 5, 11 | jca 510 | 1 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∖ cdif 3941 ⊆ wss 3944 𝒫 cpw 4604 |
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 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-in 3951 df-ss 3961 df-pw 4606 |
This theorem is referenced by: compsscnv 10396 |
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