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Mirrors > Home > ILE Home > Th. List > invdisj | Unicode version |
Description: If there is a function such that for all , then the sets for distinct are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.) |
Ref | Expression |
---|---|
invdisj | Disj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra2xy 2475 | . . 3 | |
2 | df-ral 2421 | . . . . 5 | |
3 | rsp 2480 | . . . . . . . . 9 | |
4 | eqcom 2141 | . . . . . . . . 9 | |
5 | 3, 4 | syl6ib 160 | . . . . . . . 8 |
6 | 5 | imim2i 12 | . . . . . . 7 |
7 | 6 | impd 252 | . . . . . 6 |
8 | 7 | alimi 1431 | . . . . 5 |
9 | 2, 8 | sylbi 120 | . . . 4 |
10 | mo2icl 2863 | . . . 4 | |
11 | 9, 10 | syl 14 | . . 3 |
12 | 1, 11 | alrimi 1502 | . 2 |
13 | dfdisj2 3908 | . 2 Disj | |
14 | 12, 13 | sylibr 133 | 1 Disj |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1329 wceq 1331 wcel 1480 wmo 2000 wral 2416 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-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-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rmo 2424 df-v 2688 df-disj 3907 |
This theorem is referenced by: (None) |
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