Theorem uniex2 6595

Description: The Axiom of Union using the standard abbreviation for union. Given any set , its union exists. (Contributed by NM, 4-Jun-2006.)

Assertion

Ref	Expression
uniex2

Distinct variable group:   ,

Proof of Theorem uniex2

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfun 6593	. . . 4
2		eluni 4252	. . . . . . 7
3	2	imbi1i 325	. . . . . 6
4	3	albii 1640	. . . . 5
5	4	exbii 1667	. . . 4
6	1, 5	mpbir 209	. . 3
7	6	bm1.3ii 4576	. 2
8		dfcleq 2450	. . 3
9	8	exbii 1667	. 2
10	7, 9	mpbir 209	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  U.cuni 4249

This theorem is referenced by:  uniex  6596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-uni 4250