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Mirrors > Home > MPE Home > Th. List > elpwun | Structured version Visualization version GIF version |
Description: Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.) |
Ref | Expression |
---|---|
eldifpw.1 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
elpwun | ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3455 | . 2 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) → 𝐴 ∈ V) | |
2 | elex 3455 | . . 3 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → (𝐴 ∖ 𝐶) ∈ V) | |
3 | eldifpw.1 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | difex2 7339 | . . . 4 ⊢ (𝐶 ∈ V → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V) |
6 | 2, 5 | sylibr 235 | . 2 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → 𝐴 ∈ V) |
7 | elpwg 4461 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∪ 𝐶))) | |
8 | difexg 5122 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐶) ∈ V) | |
9 | elpwg 4461 | . . . . 5 ⊢ ((𝐴 ∖ 𝐶) ∈ V → ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵)) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝐴 ∈ V → ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵)) |
11 | uncom 4050 | . . . . . 6 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
12 | 11 | sseq2i 3917 | . . . . 5 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ 𝐵)) |
13 | ssundif 4347 | . . . . 5 ⊢ (𝐴 ⊆ (𝐶 ∪ 𝐵) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) | |
14 | 12, 13 | bitri 276 | . . . 4 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) |
15 | 10, 14 | syl6rbbr 291 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)) |
16 | 7, 15 | bitrd 280 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)) |
17 | 1, 6, 16 | pm5.21nii 380 | 1 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∈ wcel 2081 Vcvv 3437 ∖ cdif 3856 ∪ cun 3857 ⊆ wss 3859 𝒫 cpw 4453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-pw 4455 df-sn 4473 df-pr 4475 df-uni 4746 |
This theorem is referenced by: pwfilem 8664 elrfi 38776 dssmapnvod 39851 |
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