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| Mirrors > Home > MPE Home > Th. List > unvdif | Structured version Visualization version GIF version | ||
| Description: The union of a class and its complement is the universe. Theorem 5.1(5) of [Stoll] p. 17. (Contributed by NM, 17-Aug-2004.) |
| Ref | Expression |
|---|---|
| unvdif | ⊢ (𝐴 ∪ (V ∖ 𝐴)) = V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfun3 4276 | . 2 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) | |
| 2 | disjdif 4472 | . . 3 ⊢ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))) = ∅ | |
| 3 | 2 | difeq2i 4123 | . 2 ⊢ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) = (V ∖ ∅) |
| 4 | dif0 4378 | . 2 ⊢ (V ∖ ∅) = V | |
| 5 | 1, 3, 4 | 3eqtri 2769 | 1 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = V |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 |
| This theorem is referenced by: undif1 4476 dfif4 4541 hashfxnn0 14376 fullfunfnv 35947 hfext 36184 |
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