Theorem univ 1964

Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235.

Assertion

Ref	Expression
univ	⊢ ∪V = V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 1884	. . 3 ⊢ ℘V = V
2	1	unieqi 1928	. 2 ⊢ ∪℘V = ∪V
3		unipw 1960	. 2 ⊢ ∪℘V = V
4	2, 3	eqtr3 1121	1 ⊢ ∪V = V

Syntax hints:   = wceq 1091  Vcvv 1348  ℘cpw 1798  ∪cuni 1919

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-uni 1920

metamath.org