Theorem elpr2elpr 3853

Description: For an element A of an unordered pair which is a subset of a given set V, there is another (maybe the same) element b of the given set V being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.)

Assertion

Ref	Expression
elpr2elpr	|- ( ( X e. V /\ Y e. V /\ A e. { X , Y } ) -> E. b e. V { X , Y } = { A , b } )

Distinct variable groups:   A, b   V, b   X, b   Y, b

Proof of Theorem elpr2elpr

Step	Hyp	Ref	Expression
1		simprr 531	. . . . . 6 |- ( ( A = X /\ ( X e. V /\ Y e. V ) ) -> Y e. V )
2		preq12 3745	. . . . . . . 8 |- ( ( A = X /\ b = Y ) -> { A , b } = { X , Y } )
3	2	eqcomd 2235	. . . . . . 7 |- ( ( A = X /\ b = Y ) -> { X , Y } = { A , b } )
4	3	adantlr 477	. . . . . 6 |- ( ( ( A = X /\ ( X e. V /\ Y e. V ) ) /\ b = Y ) -> { X , Y } = { A , b } )
5	1, 4	rspcedeq2vd 2917	. . . . 5 |- ( ( A = X /\ ( X e. V /\ Y e. V ) ) -> E. b e. V { X , Y } = { A , b } )
6	5	ex 115	. . . 4 |- ( A = X -> ( ( X e. V /\ Y e. V ) -> E. b e. V { X , Y } = { A , b } ) )
7		simprl 529	. . . . . 6 |- ( ( A = Y /\ ( X e. V /\ Y e. V ) ) -> X e. V )
8		preq12 3745	. . . . . . . 8 |- ( ( A = Y /\ b = X ) -> { A , b } = { Y , X } )
9		prcom 3742	. . . . . . . 8 |- { Y , X } = { X , Y }
10	8, 9	eqtr2di 2279	. . . . . . 7 |- ( ( A = Y /\ b = X ) -> { X , Y } = { A , b } )
11	10	adantlr 477	. . . . . 6 |- ( ( ( A = Y /\ ( X e. V /\ Y e. V ) ) /\ b = X ) -> { X , Y } = { A , b } )
12	7, 11	rspcedeq2vd 2917	. . . . 5 |- ( ( A = Y /\ ( X e. V /\ Y e. V ) ) -> E. b e. V { X , Y } = { A , b } )
13	12	ex 115	. . . 4 |- ( A = Y -> ( ( X e. V /\ Y e. V ) -> E. b e. V { X , Y } = { A , b } ) )
14	6, 13	jaoi 721	. . 3 |- ( ( A = X \/ A = Y ) -> ( ( X e. V /\ Y e. V ) -> E. b e. V { X , Y } = { A , b } ) )
15		elpri 3689	. . 3 |- ( A e. { X , Y } -> ( A = X \/ A = Y ) )
16	14, 15	syl11 31	. 2 |- ( ( X e. V /\ Y e. V ) -> ( A e. { X , Y } -> E. b e. V { X , Y } = { A , b } ) )
17	16	3impia 1224	1 |- ( ( X e. V /\ Y e. V /\ A e. { X , Y } ) -> E. b e. V { X , Y } = { A , b } )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   {cpr 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  upgredg2vtx  15940