Theorem bj-zfpair2 16853

Description: Proof of zfpair2 4345 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 16763	. . . . 5 |- BOUNDED w = x
2		ax-bdeq 16763	. . . . 5 |- BOUNDED w = y
3	1, 2	ax-bdor 16759	. . . 4 |- BOUNDED ( w = x \/ w = y )
4		ax-pr 4344	. . . 4 |- E. z A. w ( ( w = x \/ w = y ) -> w e. z )
5	3, 4	bdbm1.3ii 16834	. . 3 |- E. z A. w ( w e. z <-> ( w = x \/ w = y ) )
6		dfcleq 2232	. . . . 5 |- ( z = { x , y } <-> A. w ( w e. z <-> w e. { x , y } ) )
7		vex 2824	. . . . . . . 8 |- w e. _V
8	7	elpr 3729	. . . . . . 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 1523	. . . . 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 1658	. . 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 2831	1 |- { x , y } e. _V

Syntax hints:   <-> wb 105   \/ wo 720   A.wal 1400   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   {cpr 3709

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-pr 4344  ax-bdor 16759  ax-bdeq 16763  ax-bdsep 16827

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715

This theorem is referenced by:  bj-prexg  16854