Theorem sprvalpwn0 41733

Description: The set of all unordered pairs over a given set V, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)

Assertion

Ref	Expression
sprvalpwn0	|- ( V e. W -> (Pairs ` V ) = { p e. ( ~P V \ { (/) } ) | E. a e. V E. b e. V p = { a , b } } )

Distinct variable groups:   V, a, b, p   W, a, b, p

Proof of Theorem sprvalpwn0

Step	Hyp	Ref	Expression
1		sprvalpw 41730	. 2 |- ( V e. W -> (Pairs ` V ) = { p e. ~P V | E. a e. V E. b e. V p = { a , b } } )
2		id 22	. . . . . . . . 9 |- ( p = { a , b } -> p = { a , b } )
3		vex 3203	. . . . . . . . . . 11 |- a e. _V
4	3	prnz 4310	. . . . . . . . . 10 |- { a , b } =/= (/)
5	4	a1i 11	. . . . . . . . 9 |- ( p = { a , b } -> { a , b } =/= (/) )
6	2, 5	eqnetrd 2861	. . . . . . . 8 |- ( p = { a , b } -> p =/= (/) )
7	6	a1i 11	. . . . . . 7 |- ( ( a e. V /\ b e. V ) -> ( p = { a , b } -> p =/= (/) ) )
8	7	rexlimivv 3036	. . . . . 6 |- ( E. a e. V E. b e. V p = { a , b } -> p =/= (/) )
9	8	adantl 482	. . . . 5 |- ( ( p e. ~P V /\ E. a e. V E. b e. V p = { a , b } ) -> p =/= (/) )
10	9	pm4.71ri 665	. . . 4 |- ( ( p e. ~P V /\ E. a e. V E. b e. V p = { a , b } ) <-> ( p =/= (/) /\ ( p e. ~P V /\ E. a e. V E. b e. V p = { a , b } ) ) )
11		ancom 466	. . . . . 6 |- ( ( p =/= (/) /\ p e. ~P V ) <-> ( p e. ~P V /\ p =/= (/) ) )
12	11	anbi1i 731	. . . . 5 |- ( ( ( p =/= (/) /\ p e. ~P V ) /\ E. a e. V E. b e. V p = { a , b } ) <-> ( ( p e. ~P V /\ p =/= (/) ) /\ E. a e. V E. b e. V p = { a , b } ) )
13		anass 681	. . . . 5 |- ( ( ( p =/= (/) /\ p e. ~P V ) /\ E. a e. V E. b e. V p = { a , b } ) <-> ( p =/= (/) /\ ( p e. ~P V /\ E. a e. V E. b e. V p = { a , b } ) ) )
14		eldifsn 4317	. . . . . . 7 |- ( p e. ( ~P V \ { (/) } ) <-> ( p e. ~P V /\ p =/= (/) ) )
15	14	bicomi 214	. . . . . 6 |- ( ( p e. ~P V /\ p =/= (/) ) <-> p e. ( ~P V \ { (/) } ) )
16	15	anbi1i 731	. . . . 5 |- ( ( ( p e. ~P V /\ p =/= (/) ) /\ E. a e. V E. b e. V p = { a , b } ) <-> ( p e. ( ~P V \ { (/) } ) /\ E. a e. V E. b e. V p = { a , b } ) )
17	12, 13, 16	3bitr3i 290	. . . 4 |- ( ( p =/= (/) /\ ( p e. ~P V /\ E. a e. V E. b e. V p = { a , b } ) ) <-> ( p e. ( ~P V \ { (/) } ) /\ E. a e. V E. b e. V p = { a , b } ) )
18	10, 17	bitri 264	. . 3 |- ( ( p e. ~P V /\ E. a e. V E. b e. V p = { a , b } ) <-> ( p e. ( ~P V \ { (/) } ) /\ E. a e. V E. b e. V p = { a , b } ) )
19	18	rabbia2 3187	. 2 |- { p e. ~P V | E. a e. V E. b e. V p = { a , b } } = { p e. ( ~P V \ { (/) } ) | E. a e. V E. b e. V p = { a , b } }
20	1, 19	syl6eq 2672	1 |- ( V e. W -> (Pairs ` V ) = { p e. ( ~P V \ { (/) } ) | E. a e. V E. b e. V p = { a , b } } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179   ` cfv 5888  Pairscspr 41727

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-spr 41728

This theorem is referenced by:  sprvalpwle2  41739