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Mirrors > Home > MPE Home > Th. List > elqsi | Structured version Visualization version GIF version |
Description: Membership in a quotient set. (Contributed by NM, 23-Jul-1995.) |
Ref | Expression |
---|---|
elqsi | ⊢ (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elqsg 8806 | . 2 ⊢ (𝐵 ∈ (𝐴 / 𝑅) → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅)) | |
2 | 1 | ibi 267 | 1 ⊢ (𝐵 ∈ (𝐴 / 𝑅) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 [cec 8741 / cqs 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rex 3068 df-qs 8749 |
This theorem is referenced by: ectocld 8822 ecoptocl 8845 eroveu 8850 ghmqusker 19317 elrlocbasi 33252 nsgqusf1olem2 33421 qsidomlem2 33460 opprqusplusg 33496 opprqusmulr 33498 qsdrngi 33502 qsdrnglem2 33503 pstmxmet 33857 |
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