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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 3086 | . . . 4 | |
2 | pm5.44 910 | . . . . . 6 | |
3 | eldif 3080 | . . . . . . 7 | |
4 | 3 | imbi2i 225 | . . . . . 6 |
5 | 2, 4 | syl6bbr 197 | . . . . 5 |
6 | 5 | sps 1517 | . . . 4 |
7 | 1, 6 | sylbi 120 | . . 3 |
8 | 7 | albidv 1796 | . 2 |
9 | disj1 3413 | . 2 | |
10 | dfss2 3086 | . 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 1329 wceq 1331 wcel 1480 cdif 3068 cin 3070 wss 3071 c0 3363 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-dif 3073 df-in 3077 df-ss 3084 df-nul 3364 |
This theorem is referenced by: disj2 3418 ssdifsn 3651 structcnvcnv 11975 |
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