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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 |
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Ref | Expression |
---|---|
ecelqsi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecelqsi.1 |
. 2
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2 | ecelqsg 6614 |
. 2
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3 | 1, 2 | mpan 424 |
1
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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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-cnv 4652 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-ec 6561 df-qs 6565 |
This theorem is referenced by: ecopqsi 6616 th3q 6666 1nq 7395 addclnq 7404 mulclnq 7405 recexnq 7419 ltexnqq 7437 prarloclemarch 7447 prarloclemarch2 7448 nnnq 7451 nqnq0 7470 addnnnq0 7478 mulnnnq0 7479 addclnq0 7480 mulclnq0 7481 nqpnq0nq 7482 prarloclemlt 7522 prarloclemlo 7523 prarloclemcalc 7531 nqprm 7571 addsrpr 7774 mulsrpr 7775 0r 7779 1sr 7780 m1r 7781 addclsr 7782 mulclsr 7783 prsrcl 7813 mappsrprg 7833 suplocsrlemb 7835 pitonnlem2 7876 pitonn 7877 pitore 7879 recnnre 7880 |
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