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Mirrors > Home > ILE Home > Th. List > difexg | GIF version |
Description: Existence of a difference. (Contributed by NM, 26-May-1998.) |
Ref | Expression |
---|---|
difexg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 3108 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
2 | ssexg 3937 | . 2 ⊢ (((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∖ 𝐵) ∈ V) | |
3 | 1, 2 | mpan 415 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 Vcvv 2610 ∖ cdif 2979 ⊆ wss 2982 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2612 df-dif 2984 df-in 2988 df-ss 2995 |
This theorem is referenced by: frirrg 4133 2oconcl 6106 phplem4dom 6418 fidifsnen 6426 findcard 6444 findcard2 6445 findcard2s 6446 fisseneq 6474 |
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