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Mirrors > Home > ILE Home > Th. List > ecelqsi | GIF 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 | ⊢ 𝑅 ∈ V |
Ref | Expression |
---|---|
ecelqsi | ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecelqsi.1 | . 2 ⊢ 𝑅 ∈ V | |
2 | ecelqsg 6482 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) | |
3 | 1, 2 | mpan 420 | 1 ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 Vcvv 2686 [cec 6427 / cqs 6428 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-ec 6431 df-qs 6435 |
This theorem is referenced by: ecopqsi 6484 th3q 6534 1nq 7174 addclnq 7183 mulclnq 7184 recexnq 7198 ltexnqq 7216 prarloclemarch 7226 prarloclemarch2 7227 nnnq 7230 nqnq0 7249 addnnnq0 7257 mulnnnq0 7258 addclnq0 7259 mulclnq0 7260 nqpnq0nq 7261 prarloclemlt 7301 prarloclemlo 7302 prarloclemcalc 7310 nqprm 7350 addsrpr 7553 mulsrpr 7554 0r 7558 1sr 7559 m1r 7560 addclsr 7561 mulclsr 7562 prsrcl 7592 mappsrprg 7612 suplocsrlemb 7614 pitonnlem2 7655 pitonn 7656 pitore 7658 recnnre 7659 |
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