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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 3285 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
2 | ssexg 4168 | . 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 2164 Vcvv 2760 ∖ cdif 3150 ⊆ wss 3153 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 ax-sep 4147 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-v 2762 df-dif 3155 df-in 3159 df-ss 3166 |
This theorem is referenced by: frirrg 4381 2oconcl 6492 phplem4dom 6918 fidifsnen 6926 findcard 6944 findcard2 6945 findcard2s 6946 fisseneq 6988 difinfsn 7159 ismkvnex 7214 exmidfodomrlemim 7261 |
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