|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > qsex | Structured version Visualization version GIF version | ||
| Description: A quotient set exists. (Contributed by NM, 14-Aug-1995.) | 
| Ref | Expression | 
|---|---|
| qsex.1 | ⊢ 𝐴 ∈ V | 
| Ref | Expression | 
|---|---|
| qsex | ⊢ (𝐴 / 𝑅) ∈ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | qsex.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | qsexg 8816 | . 2 ⊢ (𝐴 ∈ V → (𝐴 / 𝑅) ∈ V) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 / 𝑅) ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 Vcvv 3479 / cqs 8745 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-rep 5278 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-rex 3070 df-v 3481 df-qs 8752 | 
| This theorem is referenced by: prjspval 42618 | 
| Copyright terms: Public domain | W3C validator |