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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 6545 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) | |
3 | 1, 2 | mpan 421 | 1 ⊢ (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2135 Vcvv 2721 [cec 6490 / cqs 6491 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-ec 6494 df-qs 6498 |
This theorem is referenced by: ecopqsi 6547 th3q 6597 1nq 7298 addclnq 7307 mulclnq 7308 recexnq 7322 ltexnqq 7340 prarloclemarch 7350 prarloclemarch2 7351 nnnq 7354 nqnq0 7373 addnnnq0 7381 mulnnnq0 7382 addclnq0 7383 mulclnq0 7384 nqpnq0nq 7385 prarloclemlt 7425 prarloclemlo 7426 prarloclemcalc 7434 nqprm 7474 addsrpr 7677 mulsrpr 7678 0r 7682 1sr 7683 m1r 7684 addclsr 7685 mulclsr 7686 prsrcl 7716 mappsrprg 7736 suplocsrlemb 7738 pitonnlem2 7779 pitonn 7780 pitore 7782 recnnre 7783 |
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