Theorem univ 5418

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 4574	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4878	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5417	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2785	1 ⊢ ∪ V = V

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3450  𝒫 cpw 4556  ∪ cuni 4866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-un 4150  df-pw 4558  df-sn 4585  df-pr 4587  df-uni 4867

This theorem is used by: (None)