Theorem bj-zfpair2 10030

Description: Proof of zfpair2 3945 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 9940	. . . . 5 |- BOUNDED w = x
2		ax-bdeq 9940	. . . . 5 |- BOUNDED w = y
3	1, 2	ax-bdor 9936	. . . 4 |- BOUNDED ( w = x \/ w = y )
4		ax-pr 3944	. . . 4 |- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
5	3, 4	bdbm1.3ii 10011	. . 3 |- E. z A. w ( w e. z <-> ( w = x \/ w = y ) )
6		dfcleq 2034	. . . . 5 |- ( z = { x , y } <-> A. w ( w e. z <-> w e. { x , y } ) )
7		vex 2560	. . . . . . . 8 |- w e. _V
8	7	elpr 3396	. . . . . . 7 |- ( w e. { x , y } <-> ( w = x \/ w = y ) )
9	8	bibi2i 216	. . . . . 6 |- ( ( w e. z <-> w e. { x , y } ) <-> ( w e. z <-> ( w = x \/ w = y ) ) )
10	9	albii 1359	. . . . 5 |- ( A. w ( w e. z <-> w e. { x , y } ) <-> A. w ( w e. z <-> ( w = x \/ w = y ) ) )
11	6, 10	bitri 173	. . . 4 |- ( z = { x , y } <-> A. w ( w e. z <-> ( w = x \/ w = y ) ) )
12	11	exbii 1496	. . 3 |- ( E. z z = { x , y } <-> E. z A. w ( w e. z <-> ( w = x \/ w = y ) ) )
13	5, 12	mpbir 134	. 2 |- E. z z = { x , y }
14	13	issetri 2564	1 |- { x , y } e. _V

Syntax hints:   <-> wb 98   \/ wo 629   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-pr 3944  ax-bdor 9936  ax-bdeq 9940  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382

This theorem is referenced by:  bj-prexg  10031