Theorem univ 4573

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	|- U. _V = _V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 3892	. . 3 |- ~P _V = _V
2	1	unieqi 3903	. 2 |- U. ~P _V = U. _V
3		unipw 4309	. 2 |- U. ~P _V = _V
4	2, 3	eqtr3i 2254	1 |- U. _V = _V

Syntax hints:   = wceq 1397   _Vcvv 2802   ~Pcpw 3652   U.cuni 3893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-uni 3894

This theorem is referenced by: (None)