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Mirrors > Home > NFE Home > Th. List > difeq1 | Unicode version |
Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
difeq1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nineq1 3234 | . . 3 &ncap ∼ &ncap ∼ | |
2 | 1 | compleqd 3245 | . 2 ∼ &ncap ∼ ∼ &ncap ∼ |
3 | df-dif 3215 | . . 3 ∼ | |
4 | df-in 3213 | . . 3 ∼ ∼ &ncap ∼ | |
5 | 3, 4 | eqtri 2373 | . 2 ∼ &ncap ∼ |
6 | df-dif 3215 | . . 3 ∼ | |
7 | df-in 3213 | . . 3 ∼ ∼ &ncap ∼ | |
8 | 6, 7 | eqtri 2373 | . 2 ∼ &ncap ∼ |
9 | 2, 5, 8 | 3eqtr4g 2410 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1642 &ncap cnin 3204 ∼ 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 |
This theorem is referenced by: symdifeq1 3248 symdifeq2 3249 difeq12 3380 difeq1i 3381 difeq1d 3384 uneqdifeq 3638 adj11 3889 |
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