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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 5122 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐵) ∈ V) | |
2 | ssun2 4070 | . . . . 5 ⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴) | |
3 | uncom 4050 | . . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐵 ∪ (𝐴 ∖ 𝐵)) | |
4 | undif2 4339 | . . . . . 6 ⊢ (𝐵 ∪ (𝐴 ∖ 𝐵)) = (𝐵 ∪ 𝐴) | |
5 | 3, 4 | eqtr2i 2820 | . . . . 5 ⊢ (𝐵 ∪ 𝐴) = ((𝐴 ∖ 𝐵) ∪ 𝐵) |
6 | 2, 5 | sseqtri 3924 | . . . 4 ⊢ 𝐴 ⊆ ((𝐴 ∖ 𝐵) ∪ 𝐵) |
7 | unexg 7329 | . . . 4 ⊢ (((𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐶) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) | |
8 | ssexg 5118 | . . . 4 ⊢ ((𝐴 ⊆ ((𝐴 ∖ 𝐵) ∪ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) → 𝐴 ∈ V) | |
9 | 6, 7, 8 | sylancr 587 | . . 3 ⊢ (((𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
10 | 9 | expcom 414 | . 2 ⊢ (𝐵 ∈ 𝐶 → ((𝐴 ∖ 𝐵) ∈ V → 𝐴 ∈ V)) |
11 | 1, 10 | impbid2 227 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐵) ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2081 Vcvv 3437 ∖ cdif 3856 ∪ cun 3857 ⊆ wss 3859 |
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-sn 4473 df-pr 4475 df-uni 4746 |
This theorem is referenced by: elpwun 7348 |
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