Theorem univ 5450

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 4904	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4920	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5449	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2763	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1542  Vcvv 3475  𝒫 cpw 4601  ∪ cuni 4907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3952  df-in 3954  df-ss 3964  df-pw 4603  df-sn 4628  df-pr 4630  df-uni 4908

This theorem is referenced by: (None)