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Mirrors > Home > MPE Home > Th. List > ecelqsg | Structured version Visualization version Unicode version |
Description: Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ecelqsg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 | |
2 | eceq1 7782 | . . . . 5 | |
3 | 2 | eqeq2d 2632 | . . . 4 |
4 | 3 | rspcev 3309 | . . 3 |
5 | 1, 4 | mpan2 707 | . 2 |
6 | ecexg 7746 | . . . 4 | |
7 | elqsg 7798 | . . . 4 | |
8 | 6, 7 | syl 17 | . . 3 |
9 | 8 | biimpar 502 | . 2 |
10 | 5, 9 | sylan2 491 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wrex 2913 cvv 3200 cec 7740 cqs 7741 |
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-8 1992 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 ax-un 6949 |
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-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ec 7744 df-qs 7748 |
This theorem is referenced by: ecelqsi 7803 qliftlem 7828 erov 7844 eroprf 7845 sylow2a 18034 sylow2blem1 18035 sylow2blem2 18036 cldsubg 21914 |
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