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Mirrors > Home > ILE Home > Th. List > disjnim | Unicode version |
Description: If a collection for is disjoint, then pairs are disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Jim Kingdon, 6-Oct-2022.) |
Ref | Expression |
---|---|
disjnim.1 |
Ref | Expression |
---|---|
disjnim | Disj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-disj 3907 | . 2 Disj | |
2 | disjnim.1 | . . . . . . 7 | |
3 | 2 | eleq2d 2209 | . . . . . 6 |
4 | 3 | rmo4 2877 | . . . . 5 |
5 | 4 | albii 1446 | . . . 4 |
6 | ralcom4 2708 | . . . 4 | |
7 | 5, 6 | bitr4i 186 | . . 3 |
8 | ralcom4 2708 | . . . . 5 | |
9 | 19.23v 1855 | . . . . . . . . 9 | |
10 | 9 | biimpi 119 | . . . . . . . 8 |
11 | 10 | necon3ad 2350 | . . . . . . 7 |
12 | notm0 3383 | . . . . . . . 8 | |
13 | elin 3259 | . . . . . . . . . 10 | |
14 | 13 | exbii 1584 | . . . . . . . . 9 |
15 | 14 | notbii 657 | . . . . . . . 8 |
16 | 12, 15 | bitr3i 185 | . . . . . . 7 |
17 | 11, 16 | syl6ibr 161 | . . . . . 6 |
18 | 17 | ralimi 2495 | . . . . 5 |
19 | 8, 18 | sylbir 134 | . . . 4 |
20 | 19 | ralimi 2495 | . . 3 |
21 | 7, 20 | sylbi 120 | . 2 |
22 | 1, 21 | sylbi 120 | 1 Disj |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wal 1329 wceq 1331 wex 1468 wcel 1480 wne 2308 wral 2416 wrmo 2419 cin 3070 c0 3363 Disj wdisj 3906 |
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-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rmo 2424 df-v 2688 df-dif 3073 df-in 3077 df-nul 3364 df-disj 3907 |
This theorem is referenced by: disjnims 3921 |
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