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Mirrors > Home > MPE Home > Th. List > difex2 | Structured version Visualization version GIF version |
Description: If the subtrahend of a class difference exists, then the minuend exists iff the difference exists. (Contributed by NM, 12-Nov-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
difex2 | ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐵) ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difexg 4841 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐵) ∈ V) | |
2 | ssun2 3810 | . . . . 5 ⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴) | |
3 | uncom 3790 | . . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐵 ∪ (𝐴 ∖ 𝐵)) | |
4 | undif2 4077 | . . . . . 6 ⊢ (𝐵 ∪ (𝐴 ∖ 𝐵)) = (𝐵 ∪ 𝐴) | |
5 | 3, 4 | eqtr2i 2674 | . . . . 5 ⊢ (𝐵 ∪ 𝐴) = ((𝐴 ∖ 𝐵) ∪ 𝐵) |
6 | 2, 5 | sseqtri 3670 | . . . 4 ⊢ 𝐴 ⊆ ((𝐴 ∖ 𝐵) ∪ 𝐵) |
7 | unexg 7001 | . . . 4 ⊢ (((𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐶) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) | |
8 | ssexg 4837 | . . . 4 ⊢ ((𝐴 ⊆ ((𝐴 ∖ 𝐵) ∪ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) → 𝐴 ∈ V) | |
9 | 6, 7, 8 | sylancr 696 | . . 3 ⊢ (((𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
10 | 9 | expcom 450 | . 2 ⊢ (𝐵 ∈ 𝐶 → ((𝐴 ∖ 𝐵) ∈ V → 𝐴 ∈ V)) |
11 | 1, 10 | impbid2 216 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐵) ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∈ wcel 2030 Vcvv 3231 ∖ cdif 3604 ∪ cun 3605 ⊆ wss 3607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-sn 4211 df-pr 4213 df-uni 4469 |
This theorem is referenced by: elpwun 7019 |
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