Theorem univ 5360

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 4833	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4849	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5359	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2769	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1543  Vcvv 3423  𝒫 cpw 4530  ∪ cuni 4836

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 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710  ax-sep 5216  ax-nul 5223  ax-pr 5346

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-v 3425  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4255  df-pw 4532  df-sn 4559  df-pr 4561  df-uni 4837

This theorem is referenced by: (None)