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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 8641 | . 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 1542 ∈ wcel 2107 ∃wrex 3072 [cec 8580 / cqs 8581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-rex 3073 df-qs 8588 |
This theorem is referenced by: ectocld 8657 ecoptocl 8680 eroveu 8685 nsgqusf1olem2 31992 qsidomlem2 32022 pstmxmet 32258 |
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