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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 3261 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
2 | ssexg 4139 | . 2 ⊢ (((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∖ 𝐵) ∈ V) | |
3 | 1, 2 | mpan 424 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 Vcvv 2737 ∖ cdif 3126 ⊆ wss 3129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4118 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 |
This theorem is referenced by: frirrg 4346 2oconcl 6433 phplem4dom 6855 fidifsnen 6863 findcard 6881 findcard2 6882 findcard2s 6883 fisseneq 6924 difinfsn 7092 ismkvnex 7146 exmidfodomrlemim 7193 |
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