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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 4735 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐵) ∈ V) | |
2 | ssun2 3739 | . . . . 5 ⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴) | |
3 | uncom 3719 | . . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐵 ∪ (𝐴 ∖ 𝐵)) | |
4 | undif2 3996 | . . . . . 6 ⊢ (𝐵 ∪ (𝐴 ∖ 𝐵)) = (𝐵 ∪ 𝐴) | |
5 | 3, 4 | eqtr2i 2633 | . . . . 5 ⊢ (𝐵 ∪ 𝐴) = ((𝐴 ∖ 𝐵) ∪ 𝐵) |
6 | 2, 5 | sseqtri 3600 | . . . 4 ⊢ 𝐴 ⊆ ((𝐴 ∖ 𝐵) ∪ 𝐵) |
7 | unexg 6857 | . . . 4 ⊢ (((𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐶) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) | |
8 | ssexg 4732 | . . . 4 ⊢ ((𝐴 ⊆ ((𝐴 ∖ 𝐵) ∪ 𝐵) ∧ ((𝐴 ∖ 𝐵) ∪ 𝐵) ∈ V) → 𝐴 ∈ V) | |
9 | 6, 7, 8 | sylancr 694 | . . 3 ⊢ (((𝐴 ∖ 𝐵) ∈ V ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V) |
10 | 9 | expcom 450 | . 2 ⊢ (𝐵 ∈ 𝐶 → ((𝐴 ∖ 𝐵) ∈ V → 𝐴 ∈ V)) |
11 | 1, 10 | impbid2 215 | 1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ V ↔ (𝐴 ∖ 𝐵) ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-pr 4128 df-uni 4373 |
This theorem is referenced by: elpwun 6869 |
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