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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 3865 | . 2 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) | |
2 | disjdif 4040 | . . 3 ⊢ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))) = ∅ | |
3 | 2 | difeq2i 3725 | . 2 ⊢ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) = (V ∖ ∅) |
4 | dif0 3950 | . 2 ⊢ (V ∖ ∅) = V | |
5 | 1, 3, 4 | 3eqtri 2648 | 1 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ∅c0 3915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 |
This theorem is referenced by: undif1 4043 dfif4 4101 hashfxnn0 13124 hashfOLD 13126 fullfunfnv 32053 hfext 32290 |
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