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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 3212 | . 2 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) → 𝐴 ∈ V) | |
2 | elex 3212 | . . 3 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → (𝐴 ∖ 𝐶) ∈ V) | |
3 | eldifpw.1 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | difex2 6969 | . . . 4 ⊢ (𝐶 ∈ V → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (𝐴 ∈ V ↔ (𝐴 ∖ 𝐶) ∈ V) |
6 | 2, 5 | sylibr 224 | . 2 ⊢ ((𝐴 ∖ 𝐶) ∈ 𝒫 𝐵 → 𝐴 ∈ V) |
7 | elpwg 4166 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∪ 𝐶))) | |
8 | difexg 4808 | . . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐶) ∈ V) | |
9 | elpwg 4166 | . . . . 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 3757 | . . . . . 6 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
12 | 11 | sseq2i 3630 | . . . . 5 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ 𝐵)) |
13 | ssundif 4052 | . . . . 5 ⊢ (𝐴 ⊆ (𝐶 ∪ 𝐵) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) | |
14 | 12, 13 | bitri 264 | . . . 4 ⊢ (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ⊆ 𝐵) |
15 | 10, 14 | syl6rbbr 279 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ⊆ (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)) |
16 | 7, 15 | bitrd 268 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵)) |
17 | 1, 6, 16 | pm5.21nii 368 | 1 ⊢ (𝐴 ∈ 𝒫 (𝐵 ∪ 𝐶) ↔ (𝐴 ∖ 𝐶) ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 𝒫 cpw 4158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-sn 4178 df-pr 4180 df-uni 4437 |
This theorem is referenced by: pwfilem 8260 elrfi 37257 dssmapnvod 38314 |
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