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| Mirrors > Home > ILE Home > Th. List > disj | Unicode version | ||
| Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.) |
| Ref | Expression |
|---|---|
| disj |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-in 3171 |
. . . 4
| |
| 2 | 1 | eqeq1i 2212 |
. . 3
|
| 3 | abeq1 2314 |
. . 3
| |
| 4 | imnan 691 |
. . . . 5
| |
| 5 | noel 3463 |
. . . . . 6
| |
| 6 | 5 | nbn 700 |
. . . . 5
|
| 7 | 4, 6 | bitr2i 185 |
. . . 4
|
| 8 | 7 | albii 1492 |
. . 3
|
| 9 | 2, 3, 8 | 3bitri 206 |
. 2
|
| 10 | df-ral 2488 |
. 2
| |
| 11 | 9, 10 | bitr4i 187 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-ext 2186 |
| This theorem depends on definitions: df-bi 117 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-v 2773 df-dif 3167 df-in 3171 df-nul 3460 |
| This theorem is referenced by: disjr 3509 disj1 3510 disjne 3513 f0rn0 5469 renfdisj 8131 fvinim0ffz 10368 xnn0nnen 10580 fxnn0nninf 10582 fprodsplitdc 11878 exmidunben 12768 dedekindeulemuub 15060 dedekindeulemlu 15064 dedekindicclemuub 15069 dedekindicclemlu 15073 ivthinclemdisj 15083 exmidsbthrlem 15923 |
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