Theorem univ 5434

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 4871	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4886	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5433	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2790	1 ⊢ ∪ V = V

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3457  𝒫 cpw 4564  ∪ cuni 4874

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-ss 3923  df-pw 4566  df-sn 4592  df-pr 4594  df-uni 4875

This theorem is used by: (None)