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Mirrors > Home > MPE Home > Th. List > relelec | Structured version Visualization version Unicode version |
Description: Membership in an equivalence class when is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
relelec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3212 | . . . 4 | |
2 | ecexr 7747 | . . . 4 | |
3 | 1, 2 | jca 554 | . . 3 |
4 | 3 | adantl 482 | . 2 |
5 | brrelex12 5155 | . . 3 | |
6 | 5 | ancomd 467 | . 2 |
7 | elecg 7785 | . 2 | |
8 | 4, 6, 7 | pm5.21nd 941 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wcel 1990 cvv 3200 class class class wbr 4653 wrel 5119 cec 7740 |
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-sbc 3436 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-res 5126 df-ima 5127 df-ec 7744 |
This theorem is referenced by: eqgid 17646 tgptsmscls 21953 pstmfval 29939 ismntop 30070 topfneec 32350 releleccnv 34021 elecres 34042 eleccnvep 34046 inecmo 34120 |
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