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Mirrors > Home > ILE 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 2934 | . . . 4 | |
2 | pm5.44 834 | . . . . . 6 | |
3 | eldif 2927 | . . . . . . 7 | |
4 | 3 | imbi2i 215 | . . . . . 6 |
5 | 2, 4 | syl6bbr 187 | . . . . 5 |
6 | 5 | sps 1430 | . . . 4 |
7 | 1, 6 | sylbi 114 | . . 3 |
8 | 7 | albidv 1705 | . 2 |
9 | disj1 3270 | . 2 | |
10 | dfss2 2934 | . 2 | |
11 | 8, 9, 10 | 3bitr4g 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wal 1241 wceq 1243 wcel 1393 cdif 2914 cin 2916 wss 2917 c0 3224 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-dif 2920 df-in 2924 df-ss 2931 df-nul 3225 |
This theorem is referenced by: disj2 3275 |
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