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Mirrors > Home > NFE Home > Th. List > reldisj | Unicode version |
Description: Two ways of saying that two classes are disjoint, using the complement of relative to a universe . (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
reldisj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3262 | . . . 4 | |
2 | pm5.44 877 | . . . . . 6 | |
3 | eldif 3221 | . . . . . . 7 | |
4 | 3 | imbi2i 303 | . . . . . 6 |
5 | 2, 4 | syl6bbr 254 | . . . . 5 |
6 | 5 | sps 1754 | . . . 4 |
7 | 1, 6 | sylbi 187 | . . 3 |
8 | 7 | albidv 1625 | . 2 |
9 | disj1 3593 | . 2 | |
10 | dfss2 3262 | . 2 | |
11 | 8, 9, 10 | 3bitr4g 279 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 176 wa 358 wal 1540 wceq 1642 wcel 1710 cdif 3206 cin 3208 wss 3257 c0 3550 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-ne 2518 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 |
This theorem is referenced by: disj2 3598 |
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