Theorem univ 5393

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 4838	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4853	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5392	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2766	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1548  Vcvv 3433  𝒫 cpw 4532  ∪ cuni 4841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-un 3890  df-ss 3902  df-pw 4534  df-sn 4559  df-pr 4561  df-uni 4842

This theorem is referenced by: (None)