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| Description: A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.) | 
| Ref | Expression | 
|---|---|
| qsexg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-qs 8751 | . 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | |
| 2 | abrexexg 7985 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} ∈ V) | |
| 3 | 1, 2 | eqeltrid 2845 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 Vcvv 3480 [cec 8743 / cqs 8744 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-rep 5279 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-v 3482 df-qs 8751 | 
| This theorem is referenced by: qsex 8816 pstmval 33894 pstmxmet 33896 imaexALTV 38331 | 
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