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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 4272 | . 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V) | |
2 | ssdif0 4203 | . 2 ⊢ (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅) | |
3 | 1, 2 | bitr3i 269 | 1 ⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1508 Vcvv 3408 ∖ cdif 3819 ⊆ wss 3822 ∅c0 4172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-v 3410 df-dif 3825 df-in 3829 df-ss 3836 df-nul 4173 |
This theorem is referenced by: setind 8968 |
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