Theorem univ 5471

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 4928	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4943	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5470	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2770	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1537  Vcvv 3488  𝒫 cpw 4622  ∪ cuni 4931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-ss 3993  df-pw 4624  df-sn 4649  df-pr 4651  df-uni 4932

This theorem is referenced by: (None)