Theorem univ 5403

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 4847	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4862	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5402	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2761	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1542  Vcvv 3429  𝒫 cpw 4541  ∪ cuni 4850

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-un 3894  df-ss 3906  df-pw 4543  df-sn 4568  df-pr 4570  df-uni 4851

This theorem is referenced by: (None)