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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 4344 | . 2 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V) | |
2 | ssdif0 4264 | . 2 ⊢ (V ⊆ 𝐴 ↔ (V ∖ 𝐴) = ∅) | |
3 | 1, 2 | bitr3i 280 | 1 ⊢ (𝐴 = V ↔ (V ∖ 𝐴) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 Vcvv 3398 ∖ cdif 3850 ⊆ wss 3853 ∅c0 4223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-dif 3856 df-in 3860 df-ss 3870 df-nul 4224 |
This theorem is referenced by: setind 9328 |
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