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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 6582 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) | |
3 | 1, 2 | mpan 424 | 1 ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 Vcvv 2737 [cec 6527 / cqs 6528 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-xp 4629 df-cnv 4631 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-ec 6531 df-qs 6535 |
This theorem is referenced by: ecopqsi 6584 th3q 6634 1nq 7356 addclnq 7365 mulclnq 7366 recexnq 7380 ltexnqq 7398 prarloclemarch 7408 prarloclemarch2 7409 nnnq 7412 nqnq0 7431 addnnnq0 7439 mulnnnq0 7440 addclnq0 7441 mulclnq0 7442 nqpnq0nq 7443 prarloclemlt 7483 prarloclemlo 7484 prarloclemcalc 7492 nqprm 7532 addsrpr 7735 mulsrpr 7736 0r 7740 1sr 7741 m1r 7742 addclsr 7743 mulclsr 7744 prsrcl 7774 mappsrprg 7794 suplocsrlemb 7796 pitonnlem2 7837 pitonn 7838 pitore 7840 recnnre 7841 |
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