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Mirrors > Home > MPE Home > Th. List > compsscnvlem | Structured version Visualization version GIF version |
Description: Lemma for compsscnv 9356. (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 479 | . . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 = (𝐴 ∖ 𝑥)) | |
2 | difss 3868 | . . . 4 ⊢ (𝐴 ∖ 𝑥) ⊆ 𝐴 | |
3 | 1, 2 | syl6eqss 3784 | . . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 ⊆ 𝐴) |
4 | selpw 4297 | . . 3 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴) | |
5 | 3, 4 | sylibr 224 | . 2 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑦 ∈ 𝒫 𝐴) |
6 | 1 | difeq2d 3859 | . . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝐴 ∖ 𝑦) = (𝐴 ∖ (𝐴 ∖ 𝑥))) |
7 | elpwi 4300 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
8 | 7 | adantr 472 | . . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑥 ⊆ 𝐴) |
9 | dfss4 3989 | . . . 4 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥) | |
10 | 8, 9 | sylib 208 | . . 3 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥) |
11 | 6, 10 | eqtr2d 2783 | . 2 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → 𝑥 = (𝐴 ∖ 𝑦)) |
12 | 5, 11 | jca 555 | 1 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 = (𝐴 ∖ 𝑥)) → (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝐴 ∖ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1620 ∈ wcel 2127 ∖ cdif 3700 ⊆ wss 3703 𝒫 cpw 4290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ral 3043 df-rab 3047 df-v 3330 df-dif 3706 df-in 3710 df-ss 3717 df-pw 4292 |
This theorem is referenced by: compsscnv 9356 |
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