Theorem univ 4491

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 3823	. . 3 |- ~P _V = _V
2	1	unieqi 3834	. 2 |- U. ~P _V = U. _V
3		unipw 4232	. 2 |- U. ~P _V = _V
4	2, 3	eqtr3i 2212	1 |- U. _V = _V

Syntax hints:   = wceq 1364   _Vcvv 2752   ~Pcpw 3590   U.cuni 3824

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-uni 3825

This theorem is referenced by: (None)