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Mirrors > Home > NFE Home > Th. List > compldif | GIF version |
Description: Complement in terms of difference. (Contributed by SF, 2-Jan-2018.) |
Ref | Expression |
---|---|
compldif | ⊢ ∼ A = (V ∖ A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dif 3215 | . 2 ⊢ (V ∖ A) = (V ∩ ∼ A) | |
2 | incom 3448 | . 2 ⊢ (V ∩ ∼ A) = ( ∼ A ∩ V) | |
3 | inv1 3577 | . 2 ⊢ ( ∼ A ∩ V) = ∼ A | |
4 | 1, 2, 3 | 3eqtrri 2378 | 1 ⊢ ∼ A = (V ∖ A) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 Vcvv 2859 ∼ ccompl 3205 ∖ cdif 3206 ∩ cin 3208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 |
This theorem is referenced by: complV 4070 |
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