Theorem univ 5451

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 4905	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4921	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5450	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2761	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1540  Vcvv 3473  𝒫 cpw 4602  ∪ cuni 4908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953  df-in 3955  df-ss 3965  df-pw 4604  df-sn 4629  df-pr 4631  df-uni 4909

This theorem is referenced by: (None)