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Mirrors > Home > ILE Home > Th. List > disjnims | Unicode version |
Description: If a collection for is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by Jim Kingdon, 7-Oct-2022.) |
Ref | Expression |
---|---|
disjnims | Disj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2299 | . . 3 | |
2 | nfcsb1v 3064 | . . 3 | |
3 | csbeq1a 3040 | . . 3 | |
4 | 1, 2, 3 | cbvdisj 3954 | . 2 Disj Disj |
5 | csbeq1 3034 | . . 3 | |
6 | 5 | disjnim 3958 | . 2 Disj |
7 | 4, 6 | sylbi 120 | 1 Disj |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1335 wne 2327 wral 2435 csb 3031 cin 3101 c0 3395 Disj wdisj 3944 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-in 3108 df-nul 3396 df-disj 3945 |
This theorem is referenced by: disji2 3960 |
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