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| Mirrors > Home > MPE Home > Th. List > univ | Structured version Visualization version GIF version | ||
| Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) | 
| Ref | Expression | 
|---|---|
| univ | ⊢ ∪ V = V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pwv 4903 | . . 3 ⊢ 𝒫 V = V | |
| 2 | 1 | unieqi 4918 | . 2 ⊢ ∪ 𝒫 V = ∪ V | 
| 3 | unipw 5454 | . 2 ⊢ ∪ 𝒫 V = V | |
| 4 | 2, 3 | eqtr3i 2766 | 1 ⊢ ∪ V = V | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 Vcvv 3479 𝒫 cpw 4599 ∪ cuni 4906 | 
| 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-sep 5295 ax-pr 5431 | 
| 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 df-pw 4601 df-sn 4626 df-pr 4628 df-uni 4907 | 
| This theorem is referenced by: (None) | 
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