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Mirrors > Home > MPE Home > Th. List > vdif0 | Structured version Visualization version GIF version |
Description: Universal class equality in terms of empty difference. (Contributed by NM, 17-Sep-2003.) |
Ref | Expression |
---|---|
vdif0 | ⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vss 4439 | . 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V) | |
2 | ssdif0 4359 | . 2 ⊢ (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅) | |
3 | 1, 2 | bitr3i 277 | 1 ⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 Vcvv 3470 ∖ cdif 3942 ⊆ wss 3945 ∅c0 4318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-dif 3948 df-in 3952 df-ss 3962 df-nul 4319 |
This theorem is referenced by: setind 9751 |
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