Theorem univ 4392

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 3730	. . 3 |- ~P _V = _V
2	1	unieqi 3741	. 2 |- U. ~P _V = U. _V
3		unipw 4134	. 2 |- U. ~P _V = _V
4	2, 3	eqtr3i 2160	1 |- U. _V = _V

Syntax hints:   = wceq 1331   _Vcvv 2681   ~Pcpw 3505   U.cuni 3731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-uni 3732

This theorem is referenced by: (None)