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Mirrors > Home > ILE Home > Th. List > vdif0im | GIF version |
Description: Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.) |
Ref | Expression |
---|---|
vdif0im | ⊢ (𝐴 = V → (V ∖ 𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vss 3470 | . 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V) | |
2 | ssdif0im 3487 | . 2 ⊢ (V ⊆ 𝐴 → (V ∖ 𝐴) = ∅) | |
3 | 1, 2 | sylbir 135 | 1 ⊢ (𝐴 = V → (V ∖ 𝐴) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 Vcvv 2737 ∖ cdif 3126 ⊆ wss 3129 ∅c0 3422 |
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 |
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 df-nul 3423 |
This theorem is referenced by: (None) |
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