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Mirrors > Home > ILE Home > Th. List > xpider | Unicode version |
Description: A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
xpider |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 4618 | . 2 | |
2 | dmxpid 4730 | . 2 | |
3 | cnvxp 4927 | . . 3 | |
4 | xpidtr 4899 | . . 3 | |
5 | uneq1 3193 | . . . 4 | |
6 | unss2 3217 | . . . 4 | |
7 | unidm 3189 | . . . . 5 | |
8 | eqtr 2135 | . . . . . 6 | |
9 | sseq2 3091 | . . . . . . 7 | |
10 | 9 | biimpd 143 | . . . . . 6 |
11 | 8, 10 | syl 14 | . . . . 5 |
12 | 7, 11 | mpan2 421 | . . . 4 |
13 | 5, 6, 12 | syl2im 38 | . . 3 |
14 | 3, 4, 13 | mp2 16 | . 2 |
15 | df-er 6397 | . 2 | |
16 | 1, 2, 14, 15 | mpbir3an 1148 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 cun 3039 wss 3041 cxp 4507 ccnv 4508 cdm 4509 ccom 4513 wrel 4514 wer 6394 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-er 6397 |
This theorem is referenced by: (None) |
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