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Mirrors > Home > MPE Home > Th. List > erssxp | Structured version Visualization version Unicode version |
Description: An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
erssxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | errel 7751 | . . 3 | |
2 | relssdmrn 5656 | . . 3 | |
3 | 1, 2 | syl 17 | . 2 |
4 | erdm 7752 | . . 3 | |
5 | errn 7764 | . . 3 | |
6 | 4, 5 | xpeq12d 5140 | . 2 |
7 | 3, 6 | sseqtrd 3641 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wss 3574 cxp 5112 cdm 5114 crn 5115 wrel 5119 wer 7739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-er 7742 |
This theorem is referenced by: erex 7766 riiner 7820 efgval 18130 qtophaus 29903 |
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