Theorem univ 5455

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 4903	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4918	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5454	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2766	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1539  Vcvv 3479  𝒫 cpw 4599  ∪ cuni 4906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955  df-ss 3967  df-pw 4601  df-sn 4626  df-pr 4628  df-uni 4907

This theorem is referenced by: (None)