Theorem univ 5406

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 4862	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4877	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5405	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2762	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1542  Vcvv 3442  𝒫 cpw 4556  ∪ cuni 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-ss 3920  df-pw 4558  df-sn 4583  df-pr 4585  df-uni 4866

This theorem is referenced by: (None)