Theorem bj-zfpair2 14598

Description: Proof of zfpair2 4210 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-zfpair2	|- { x , y } e. _V

Proof of Theorem bj-zfpair2

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-bdeq 14508	. . . . 5 |- BOUNDED w = x
2		ax-bdeq 14508	. . . . 5 |- BOUNDED w = y
3	1, 2	ax-bdor 14504	. . . 4 |- BOUNDED ( w = x \/ w = y )
4		ax-pr 4209	. . . 4 |- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
5	3, 4	bdbm1.3ii 14579	. . 3 |- E. z A. w ( w e. z <-> ( w = x \/ w = y ) )
6		dfcleq 2171	. . . . 5 |- ( z = { x , y } <-> A. w ( w e. z <-> w e. { x , y } ) )
7		vex 2740	. . . . . . . 8 |- w e. _V
8	7	elpr 3613	. . . . . . 7 |- ( w e. { x , y } <-> ( w = x \/ w = y ) )
9	8	bibi2i 227	. . . . . 6 |- ( ( w e. z <-> w e. { x , y } ) <-> ( w e. z <-> ( w = x \/ w = y ) ) )
10	9	albii 1470	. . . . 5 |- ( A. w ( w e. z <-> w e. { x , y } ) <-> A. w ( w e. z <-> ( w = x \/ w = y ) ) )
11	6, 10	bitri 184	. . . 4 |- ( z = { x , y } <-> A. w ( w e. z <-> ( w = x \/ w = y ) ) )
12	11	exbii 1605	. . 3 |- ( E. z z = { x , y } <-> E. z A. w ( w e. z <-> ( w = x \/ w = y ) ) )
13	5, 12	mpbir 146	. 2 |- E. z z = { x , y }
14	13	issetri 2746	1 |- { x , y } e. _V

Syntax hints:   <-> wb 105   \/ wo 708   A.wal 1351   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2737   {cpr 3593

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-pr 4209  ax-bdor 14504  ax-bdeq 14508  ax-bdsep 14572

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599

This theorem is referenced by:  bj-prexg  14599