![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4905 | . . 3 ⊢ 𝒫 V = V | |
2 | 1 | unieqi 4921 | . 2 ⊢ ∪ 𝒫 V = ∪ V |
3 | unipw 5450 | . 2 ⊢ ∪ 𝒫 V = V | |
4 | 2, 3 | eqtr3i 2761 | 1 ⊢ ∪ V = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 Vcvv 3473 𝒫 cpw 4602 ∪ cuni 4908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-un 3953 df-in 3955 df-ss 3965 df-pw 4604 df-sn 4629 df-pr 4631 df-uni 4909 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |