Theorem univ 5415

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 4859	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4874	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5414	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2786	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1559  Vcvv 3453  𝒫 cpw 4552  ∪ cuni 4862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3907  df-ss 3919  df-pw 4554  df-sn 4580  df-pr 4582  df-uni 4863

This theorem is referenced by: (None)