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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 3405 | . 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V) | |
2 | ssdif0im 3422 | . 2 ⊢ (V ⊆ 𝐴 → (V ∖ 𝐴) = ∅) | |
3 | 1, 2 | sylbir 134 | 1 ⊢ (𝐴 = V → (V ∖ 𝐴) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Vcvv 2681 ∖ cdif 3063 ⊆ wss 3066 ∅c0 3358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-in 3072 df-ss 3079 df-nul 3359 |
This theorem is referenced by: (None) |
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