Theorem univ 5024

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 4541	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 4553	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 5023	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2748	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1596  Vcvv 3304  𝒫 cpw 4266  ∪ cuni 4544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rex 3020  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-pw 4268  df-sn 4286  df-pr 4288  df-uni 4545

This theorem is referenced by: (None)