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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 3126 | . . . 4 | |
2 | pm5.44 915 | . . . . . 6 | |
3 | eldif 3120 | . . . . . . 7 | |
4 | 3 | imbi2i 225 | . . . . . 6 |
5 | 2, 4 | bitr4di 197 | . . . . 5 |
6 | 5 | sps 1524 | . . . 4 |
7 | 1, 6 | sylbi 120 | . . 3 |
8 | 7 | albidv 1811 | . 2 |
9 | disj1 3454 | . 2 | |
10 | dfss2 3126 | . 2 | |
11 | 8, 9, 10 | 3bitr4g 222 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wal 1340 wceq 1342 wcel 2135 cdif 3108 cin 3110 wss 3111 c0 3404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-dif 3113 df-in 3117 df-ss 3124 df-nul 3405 |
This theorem is referenced by: disj2 3459 ssdifsn 3698 structcnvcnv 12353 |
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