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Mirrors > Home > ILE Home > Th. List > disjxp1 | Unicode version |
Description: The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
disjxp1.1 | Disj |
Ref | Expression |
---|---|
disjxp1 | Disj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xp1st 6063 | . . . . . . 7 | |
2 | xp1st 6063 | . . . . . . 7 | |
3 | disjxp1.1 | . . . . . . . . . . . 12 Disj | |
4 | df-disj 3907 | . . . . . . . . . . . 12 Disj | |
5 | 3, 4 | sylib 121 | . . . . . . . . . . 11 |
6 | 1stexg 6065 | . . . . . . . . . . . . 13 | |
7 | 6 | elv 2690 | . . . . . . . . . . . 12 |
8 | eleq1 2202 | . . . . . . . . . . . . 13 | |
9 | 8 | rmobidv 2619 | . . . . . . . . . . . 12 |
10 | 7, 9 | spcv 2779 | . . . . . . . . . . 11 |
11 | 5, 10 | syl 14 | . . . . . . . . . 10 |
12 | nfcv 2281 | . . . . . . . . . . 11 | |
13 | nfcv 2281 | . . . . . . . . . . 11 | |
14 | nfcsb1v 3035 | . . . . . . . . . . . 12 | |
15 | 14 | nfel2 2294 | . . . . . . . . . . 11 |
16 | csbeq1a 3012 | . . . . . . . . . . . 12 | |
17 | 16 | eleq2d 2209 | . . . . . . . . . . 11 |
18 | 12, 13, 15, 17 | rmo4f 2882 | . . . . . . . . . 10 |
19 | 11, 18 | sylib 121 | . . . . . . . . 9 |
20 | 19 | r19.21bi 2520 | . . . . . . . 8 |
21 | 20 | r19.21bi 2520 | . . . . . . 7 |
22 | 1, 2, 21 | syl2ani 405 | . . . . . 6 |
23 | 22 | ralrimiva 2505 | . . . . 5 |
24 | 23 | ralrimiva 2505 | . . . 4 |
25 | nfcsb1v 3035 | . . . . . . 7 | |
26 | 14, 25 | nfxp 4566 | . . . . . 6 |
27 | 26 | nfel2 2294 | . . . . 5 |
28 | csbeq1a 3012 | . . . . . . 7 | |
29 | 16, 28 | xpeq12d 4564 | . . . . . 6 |
30 | 29 | eleq2d 2209 | . . . . 5 |
31 | 12, 13, 27, 30 | rmo4f 2882 | . . . 4 |
32 | 24, 31 | sylibr 133 | . . 3 |
33 | 32 | alrimiv 1846 | . 2 |
34 | df-disj 3907 | . 2 Disj | |
35 | 33, 34 | sylibr 133 | 1 Disj |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1329 wceq 1331 wcel 1480 wral 2416 wrmo 2419 cvv 2686 csb 3003 Disj wdisj 3906 cxp 4537 cfv 5123 c1st 6036 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 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-rex 2422 df-rmo 2424 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-disj 3907 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 df-1st 6038 |
This theorem is referenced by: disjsnxp 6134 |
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