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| Mirrors > Home > ILE Home > Th. List > ecelqsi | Unicode version | ||
| Description: Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) |
| Ref | Expression |
|---|---|
| ecelqsi.1 |
|
| Ref | Expression |
|---|---|
| ecelqsi |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ecelqsi.1 |
. 2
| |
| 2 | ecelqsg 6700 |
. 2
| |
| 3 | 1, 2 | mpan 424 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4179 ax-pow 4235 ax-pr 4270 ax-un 4499 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ral 2491 df-rex 2492 df-v 2779 df-un 3179 df-in 3181 df-ss 3188 df-pw 3629 df-sn 3650 df-pr 3651 df-op 3653 df-uni 3866 df-br 4061 df-opab 4123 df-xp 4700 df-cnv 4702 df-dm 4704 df-rn 4705 df-res 4706 df-ima 4707 df-ec 6647 df-qs 6651 |
| This theorem is referenced by: ecopqsi 6702 th3q 6752 1nq 7516 addclnq 7525 mulclnq 7526 recexnq 7540 ltexnqq 7558 prarloclemarch 7568 prarloclemarch2 7569 nnnq 7572 nqnq0 7591 addnnnq0 7599 mulnnnq0 7600 addclnq0 7601 mulclnq0 7602 nqpnq0nq 7603 prarloclemlt 7643 prarloclemlo 7644 prarloclemcalc 7652 nqprm 7692 addsrpr 7895 mulsrpr 7896 0r 7900 1sr 7901 m1r 7902 addclsr 7903 mulclsr 7904 prsrcl 7934 mappsrprg 7954 suplocsrlemb 7956 pitonnlem2 7997 pitonn 7998 pitore 8000 recnnre 8001 |
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