Theorem elpr2elpr 4398

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 796	. . . . . 6 |- ( ( A = X /\ ( X e. V /\ Y e. V ) ) -> Y e. V )
2		preq2 4269	. . . . . . . 8 |- ( b = Y -> { A , b } = { A , Y } )
3	2	eqeq2d 2632	. . . . . . 7 |- ( b = Y -> ( { X , Y } = { A , b } <-> { X , Y } = { A , Y } ) )
4	3	adantl 482	. . . . . 6 |- ( ( ( A = X /\ ( X e. V /\ Y e. V ) ) /\ b = Y ) -> ( { X , Y } = { A , b } <-> { X , Y } = { A , Y } ) )
5		preq1 4268	. . . . . . . 8 |- ( X = A -> { X , Y } = { A , Y } )
6	5	eqcoms 2630	. . . . . . 7 |- ( A = X -> { X , Y } = { A , Y } )
7	6	adantr 481	. . . . . 6 |- ( ( A = X /\ ( X e. V /\ Y e. V ) ) -> { X , Y } = { A , Y } )
8	1, 4, 7	rspcedvd 3317	. . . . 5 |- ( ( A = X /\ ( X e. V /\ Y e. V ) ) -> E. b e. V { X , Y } = { A , b } )
9	8	ex 450	. . . 4 |- ( A = X -> ( ( X e. V /\ Y e. V ) -> E. b e. V { X , Y } = { A , b } ) )
10		simprl 794	. . . . . 6 |- ( ( A = Y /\ ( X e. V /\ Y e. V ) ) -> X e. V )
11		preq2 4269	. . . . . . . 8 |- ( b = X -> { A , b } = { A , X } )
12	11	eqeq2d 2632	. . . . . . 7 |- ( b = X -> ( { X , Y } = { A , b } <-> { X , Y } = { A , X } ) )
13	12	adantl 482	. . . . . 6 |- ( ( ( A = Y /\ ( X e. V /\ Y e. V ) ) /\ b = X ) -> ( { X , Y } = { A , b } <-> { X , Y } = { A , X } ) )
14		preq2 4269	. . . . . . . . 9 |- ( Y = A -> { X , Y } = { X , A } )
15	14	eqcoms 2630	. . . . . . . 8 |- ( A = Y -> { X , Y } = { X , A } )
16		prcom 4267	. . . . . . . 8 |- { X , A } = { A , X }
17	15, 16	syl6eq 2672	. . . . . . 7 |- ( A = Y -> { X , Y } = { A , X } )
18	17	adantr 481	. . . . . 6 |- ( ( A = Y /\ ( X e. V /\ Y e. V ) ) -> { X , Y } = { A , X } )
19	10, 13, 18	rspcedvd 3317	. . . . 5 |- ( ( A = Y /\ ( X e. V /\ Y e. V ) ) -> E. b e. V { X , Y } = { A , b } )
20	19	ex 450	. . . 4 |- ( A = Y -> ( ( X e. V /\ Y e. V ) -> E. b e. V { X , Y } = { A , b } ) )
21	9, 20	jaoi 394	. . 3 |- ( ( A = X \/ A = Y ) -> ( ( X e. V /\ Y e. V ) -> E. b e. V { X , Y } = { A , b } ) )
22		elpri 4197	. . 3 |- ( A e. { X , Y } -> ( A = X \/ A = Y ) )
23	21, 22	syl11 33	. 2 |- ( ( X e. V /\ Y e. V ) -> ( A e. { X , Y } -> E. b e. V { X , Y } = { A , b } ) )
24	23	3impia 1261	1 |- ( ( X e. V /\ Y e. V /\ A e. { X , Y } ) -> E. b e. V { X , Y } = { A , b } )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180

This theorem is referenced by:  upgredg2vtx  26036