Theorem univ 5312

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 4795	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4811	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5311	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2783	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1538  Vcvv 3409  𝒫 cpw 4494  ∪ cuni 4798

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-pw 4496  df-sn 4523  df-pr 4525  df-uni 4799

This theorem is referenced by: (None)