Theorem elsnxpOLD 5678

Description: Obsolete proof of elsnxp 5677 as of 14-Jul-2021. (Contributed by Thierry Arnoux, 10-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
elsnxpOLD	|- ( X e. V -> ( Z e. ( { X } X. A ) <-> E. y e. A Z = <. X , y >. ) )

Distinct variable groups:   y, A   y, V   y, X   y, Z

Proof of Theorem elsnxpOLD

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 5131	. . . 4 |- ( Z e. ( { X } X. A ) <-> E. x E. y ( Z = <. x , y >. /\ ( x e. { X } /\ y e. A ) ) )
2		df-rex 2918	. . . . . . . 8 |- ( E. y e. A ( x e. { X } /\ Z = <. x , y >. ) <-> E. y ( y e. A /\ ( x e. { X } /\ Z = <. x , y >. ) ) )
3		an13 840	. . . . . . . . 9 |- ( ( Z = <. x , y >. /\ ( x e. { X } /\ y e. A ) ) <-> ( y e. A /\ ( x e. { X } /\ Z = <. x , y >. ) ) )
4	3	exbii 1774	. . . . . . . 8 |- ( E. y ( Z = <. x , y >. /\ ( x e. { X } /\ y e. A ) ) <-> E. y ( y e. A /\ ( x e. { X } /\ Z = <. x , y >. ) ) )
5	2, 4	bitr4i 267	. . . . . . 7 |- ( E. y e. A ( x e. { X } /\ Z = <. x , y >. ) <-> E. y ( Z = <. x , y >. /\ ( x e. { X } /\ y e. A ) ) )
6		velsn 4193	. . . . . . . . . 10 |- ( x e. { X } <-> x = X )
7	6	anbi1i 731	. . . . . . . . 9 |- ( ( x e. { X } /\ Z = <. x , y >. ) <-> ( x = X /\ Z = <. x , y >. ) )
8		simpr 477	. . . . . . . . . 10 |- ( ( x = X /\ Z = <. x , y >. ) -> Z = <. x , y >. )
9		opeq1 4402	. . . . . . . . . . 11 |- ( x = X -> <. x , y >. = <. X , y >. )
10	9	adantr 481	. . . . . . . . . 10 |- ( ( x = X /\ Z = <. x , y >. ) -> <. x , y >. = <. X , y >. )
11	8, 10	eqtrd 2656	. . . . . . . . 9 |- ( ( x = X /\ Z = <. x , y >. ) -> Z = <. X , y >. )
12	7, 11	sylbi 207	. . . . . . . 8 |- ( ( x e. { X } /\ Z = <. x , y >. ) -> Z = <. X , y >. )
13	12	reximi 3011	. . . . . . 7 |- ( E. y e. A ( x e. { X } /\ Z = <. x , y >. ) -> E. y e. A Z = <. X , y >. )
14	5, 13	sylbir 225	. . . . . 6 |- ( E. y ( Z = <. x , y >. /\ ( x e. { X } /\ y e. A ) ) -> E. y e. A Z = <. X , y >. )
15	14	eximi 1762	. . . . 5 |- ( E. x E. y ( Z = <. x , y >. /\ ( x e. { X } /\ y e. A ) ) -> E. x E. y e. A Z = <. X , y >. )
16		19.9v 1896	. . . . 5 |- ( E. x E. y e. A Z = <. X , y >. <-> E. y e. A Z = <. X , y >. )
17	15, 16	sylib 208	. . . 4 |- ( E. x E. y ( Z = <. x , y >. /\ ( x e. { X } /\ y e. A ) ) -> E. y e. A Z = <. X , y >. )
18	1, 17	sylbi 207	. . 3 |- ( Z e. ( { X } X. A ) -> E. y e. A Z = <. X , y >. )
19	18	adantl 482	. 2 |- ( ( X e. V /\ Z e. ( { X } X. A ) ) -> E. y e. A Z = <. X , y >. )
20		nfv 1843	. . . 4 |- F/ y X e. V
21		nfre1 3005	. . . 4 |- F/ y E. y e. A Z = <. X , y >.
22	20, 21	nfan 1828	. . 3 |- F/ y ( X e. V /\ E. y e. A Z = <. X , y >. )
23		simpr 477	. . . . 5 |- ( ( ( X e. V /\ y e. A ) /\ Z = <. X , y >. ) -> Z = <. X , y >. )
24		snidg 4206	. . . . . . . 8 |- ( X e. V -> X e. { X } )
25	24	adantr 481	. . . . . . 7 |- ( ( X e. V /\ y e. A ) -> X e. { X } )
26		simpr 477	. . . . . . 7 |- ( ( X e. V /\ y e. A ) -> y e. A )
27		opelxp 5146	. . . . . . . 8 |- ( <. X , y >. e. ( { X } X. A ) <-> ( X e. { X } /\ y e. A ) )
28	27	biimpri 218	. . . . . . 7 |- ( ( X e. { X } /\ y e. A ) -> <. X , y >. e. ( { X } X. A ) )
29	25, 26, 28	syl2anc 693	. . . . . 6 |- ( ( X e. V /\ y e. A ) -> <. X , y >. e. ( { X } X. A ) )
30	29	adantr 481	. . . . 5 |- ( ( ( X e. V /\ y e. A ) /\ Z = <. X , y >. ) -> <. X , y >. e. ( { X } X. A ) )
31	23, 30	eqeltrd 2701	. . . 4 |- ( ( ( X e. V /\ y e. A ) /\ Z = <. X , y >. ) -> Z e. ( { X } X. A ) )
32	31	adantllr 755	. . 3 |- ( ( ( ( X e. V /\ E. y e. A Z = <. X , y >. ) /\ y e. A ) /\ Z = <. X , y >. ) -> Z e. ( { X } X. A ) )
33		simpr 477	. . 3 |- ( ( X e. V /\ E. y e. A Z = <. X , y >. ) -> E. y e. A Z = <. X , y >. )
34	22, 32, 33	r19.29af 3076	. 2 |- ( ( X e. V /\ E. y e. A Z = <. X , y >. ) -> Z e. ( { X } X. A ) )
35	19, 34	impbida 877	1 |- ( X e. V -> ( Z e. ( { X } X. A ) <-> E. y e. A Z = <. X , y >. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   {csn 4177   <.cop 4183   X. cxp 5112

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120

This theorem is referenced by: (None)