Theorem bj-zfpair2 13456

Description: Proof of zfpair2 4170 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 13366	. . . . 5 |- BOUNDED w = x
2		ax-bdeq 13366	. . . . 5 |- BOUNDED w = y
3	1, 2	ax-bdor 13362	. . . 4 |- BOUNDED ( w = x \/ w = y )
4		ax-pr 4169	. . . 4 |- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
5	3, 4	bdbm1.3ii 13437	. . 3 |- E. z A. w ( w e. z <-> ( w = x \/ w = y ) )
6		dfcleq 2151	. . . . 5 |- ( z = { x , y } <-> A. w ( w e. z <-> w e. { x , y } ) )
7		vex 2715	. . . . . . . 8 |- w e. _V
8	7	elpr 3581	. . . . . . 7 |- ( w e. { x , y } <-> ( w = x \/ w = y ) )
9	8	bibi2i 226	. . . . . 6 |- ( ( w e. z <-> w e. { x , y } ) <-> ( w e. z <-> ( w = x \/ w = y ) ) )
10	9	albii 1450	. . . . 5 |- ( A. w ( w e. z <-> w e. { x , y } ) <-> A. w ( w e. z <-> ( w = x \/ w = y ) ) )
11	6, 10	bitri 183	. . . 4 |- ( z = { x , y } <-> A. w ( w e. z <-> ( w = x \/ w = y ) ) )
12	11	exbii 1585	. . 3 |- ( E. z z = { x , y } <-> E. z A. w ( w e. z <-> ( w = x \/ w = y ) ) )
13	5, 12	mpbir 145	. 2 |- E. z z = { x , y }
14	13	issetri 2721	1 |- { x , y } e. _V

Syntax hints:   <-> wb 104   \/ wo 698   A.wal 1333   = wceq 1335   E.wex 1472   e. wcel 2128   _Vcvv 2712   {cpr 3561

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-pr 4169  ax-bdor 13362  ax-bdeq 13366  ax-bdsep 13430

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567

This theorem is referenced by:  bj-prexg  13457