|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > unv | Structured version Visualization version GIF version | ||
| Description: The union of a class with the universal class is the universal class. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.) | 
| Ref | Expression | 
|---|---|
| unv | ⊢ (𝐴 ∪ V) = V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssv 4007 | . 2 ⊢ (𝐴 ∪ V) ⊆ V | |
| 2 | ssun2 4178 | . 2 ⊢ V ⊆ (𝐴 ∪ V) | |
| 3 | 1, 2 | eqssi 3999 | 1 ⊢ (𝐴 ∪ V) = V | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 Vcvv 3479 ∪ cun 3948 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 df-ss 3967 | 
| This theorem is referenced by: oev2 8562 | 
| Copyright terms: Public domain | W3C validator |