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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 4648 | . 2 | |
2 | dmxpid 4760 | . 2 | |
3 | cnvxp 4957 | . . 3 | |
4 | xpidtr 4929 | . . 3 | |
5 | uneq1 3223 | . . . 4 | |
6 | unss2 3247 | . . . 4 | |
7 | unidm 3219 | . . . . 5 | |
8 | eqtr 2157 | . . . . . 6 | |
9 | sseq2 3121 | . . . . . . 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 6429 | . 2 | |
16 | 1, 2, 14, 15 | mpbir3an 1163 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 cun 3069 wss 3071 cxp 4537 ccnv 4538 cdm 4539 ccom 4543 wrel 4544 wer 6426 |
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-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 |
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-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-er 6429 |
This theorem is referenced by: (None) |
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