Theorem spr0nelg 41726

Description: The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.)

Assertion

Ref	Expression
spr0nelg	|- (/) e/ { p | E. a E. b p = { a , b } }

Distinct variable groups:   p, a   p, b

Proof of Theorem spr0nelg

Step	Hyp	Ref	Expression
1		ianor 509	. . . . . 6 |- ( -. ( p = (/) /\ E. a E. b p = { a , b } ) <-> ( -. p = (/) \/ -. E. a E. b p = { a , b } ) )
2	1	bicomi 214	. . . . 5 |- ( ( -. p = (/) \/ -. E. a E. b p = { a , b } ) <-> -. ( p = (/) /\ E. a E. b p = { a , b } ) )
3	2	albii 1747	. . . 4 |- ( A. p ( -. p = (/) \/ -. E. a E. b p = { a , b } ) <-> A. p -. ( p = (/) /\ E. a E. b p = { a , b } ) )
4		alnex 1706	. . . 4 |- ( A. p -. ( p = (/) /\ E. a E. b p = { a , b } ) <-> -. E. p ( p = (/) /\ E. a E. b p = { a , b } ) )
5	3, 4	bitri 264	. . 3 |- ( A. p ( -. p = (/) \/ -. E. a E. b p = { a , b } ) <-> -. E. p ( p = (/) /\ E. a E. b p = { a , b } ) )
6		vex 3203	. . . . . . . . 9 |- a e. _V
7	6	prnz 4310	. . . . . . . 8 |- { a , b } =/= (/)
8	7	nesymi 2851	. . . . . . 7 |- -. (/) = { a , b }
9		eqeq1 2626	. . . . . . 7 |- ( p = (/) -> ( p = { a , b } <-> (/) = { a , b } ) )
10	8, 9	mtbiri 317	. . . . . 6 |- ( p = (/) -> -. p = { a , b } )
11	10	alrimivv 1856	. . . . 5 |- ( p = (/) -> A. a A. b -. p = { a , b } )
12		2nexaln 1757	. . . . 5 |- ( -. E. a E. b p = { a , b } <-> A. a A. b -. p = { a , b } )
13	11, 12	sylibr 224	. . . 4 |- ( p = (/) -> -. E. a E. b p = { a , b } )
14	13	imori 429	. . 3 |- ( -. p = (/) \/ -. E. a E. b p = { a , b } )
15	5, 14	mpgbi 1725	. 2 |- -. E. p ( p = (/) /\ E. a E. b p = { a , b } )
16		df-nel 2898	. . 3 |- ( (/) e/ { p | E. a E. b p = { a , b } } <-> -. (/) e. { p | E. a E. b p = { a , b } } )
17		clelab 2748	. . 3 |- ( (/) e. { p | E. a E. b p = { a , b } } <-> E. p ( p = (/) /\ E. a E. b p = { a , b } ) )
18	16, 17	xchbinx 324	. 2 |- ( (/) e/ { p | E. a E. b p = { a , b } } <-> -. E. p ( p = (/) /\ E. a E. b p = { a , b } ) )
19	15, 18	mpbir 221	1 |- (/) e/ { p | E. a E. b p = { a , b } }

Syntax hints:   -. wn 3   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   e/ wnel 2897   (/)c0 3915   {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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by:  spr0el  41732