Theorem univ 5422

Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
univ	⊢ ∪ V = V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 4864	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4879	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5421	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2790	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1563  Vcvv 3457  𝒫 cpw 4558  ∪ cuni 4867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3912  df-ss 3924  df-pw 4560  df-sn 4586  df-pr 4588  df-uni 4868

This theorem is referenced by: (None)