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 2117 | . . 3 | |
2 | eceq1 6432 | . . . . 5 | |
3 | 2 | eqeq2d 2129 | . . . 4 |
4 | 3 | rspcev 2763 | . . 3 |
5 | 1, 4 | mpan2 421 | . 2 |
6 | ecexg 6401 | . . . 4 | |
7 | elqsg 6447 | . . . 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 1316 wcel 1465 wrex 2394 cvv 2660 cec 6395 cqs 6396 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-ec 6399 df-qs 6403 |
This theorem is referenced by: ecelqsi 6451 qliftlem 6475 eroprf 6490 |
Copyright terms: Public domain | W3C validator |