Theorem univ 5441

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 4896	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4911	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5440	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2754	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1533  Vcvv 3466  𝒫 cpw 4594  ∪ cuni 4899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3945  df-in 3947  df-ss 3957  df-pw 4596  df-sn 4621  df-pr 4623  df-uni 4900

This theorem is referenced by: (None)