Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ecelqsg | 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 2164 | . . 3 | |
2 | eceq1 6528 | . . . . 5 | |
3 | 2 | eqeq2d 2176 | . . . 4 |
4 | 3 | rspcev 2826 | . . 3 |
5 | 1, 4 | mpan2 422 | . 2 |
6 | ecexg 6497 | . . . 4 | |
7 | elqsg 6543 | . . . 4 | |
8 | 6, 7 | syl 14 | . . 3 |
9 | 8 | biimpar 295 | . 2 |
10 | 5, 9 | sylan2 284 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wrex 2443 cvv 2722 cec 6491 cqs 6492 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-xp 4605 df-cnv 4607 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-ec 6495 df-qs 6499 |
This theorem is referenced by: ecelqsi 6547 qliftlem 6571 eroprf 6586 |
Copyright terms: Public domain | W3C validator |