Theorem bj-zfpair2 15402

Description: Proof of zfpair2 4239 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 15312	. . . . 5 |- BOUNDED w = x
2		ax-bdeq 15312	. . . . 5 |- BOUNDED w = y
3	1, 2	ax-bdor 15308	. . . 4 |- BOUNDED ( w = x \/ w = y )
4		ax-pr 4238	. . . 4 |- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
5	3, 4	bdbm1.3ii 15383	. . 3 |- E. z A. w ( w e. z <-> ( w = x \/ w = y ) )
6		dfcleq 2187	. . . . 5 |- ( z = { x , y } <-> A. w ( w e. z <-> w e. { x , y } ) )
7		vex 2763	. . . . . . . 8 |- w e. _V
8	7	elpr 3639	. . . . . . 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 1481	. . . . 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 1616	. . 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 2769	1 |- { x , y } e. _V

Syntax hints:   <-> wb 105   \/ wo 709   A.wal 1362   = wceq 1364   E.wex 1503   e. wcel 2164   _Vcvv 2760   {cpr 3619

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-pr 4238  ax-bdor 15308  ax-bdeq 15312  ax-bdsep 15376

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625

This theorem is referenced by:  bj-prexg  15403