Theorem univ 5200

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 4709	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4721	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5199	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2804	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1507  Vcvv 3415  𝒫 cpw 4422  ∪ cuni 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rex 3094  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-pw 4424  df-sn 4442  df-pr 4444  df-uni 4713

This theorem is referenced by: (None)