Theorem univ 4511

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 3838	. . 3 |- ~P _V = _V
2	1	unieqi 3849	. 2 |- U. ~P _V = U. _V
3		unipw 4250	. 2 |- U. ~P _V = _V
4	2, 3	eqtr3i 2219	1 |- U. _V = _V

Syntax hints:   = wceq 1364   _Vcvv 2763   ~Pcpw 3605   U.cuni 3839

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-uni 3840

This theorem is referenced by: (None)