Theorem elnonrel 37891

Description: Only an ordered pair where not both entries are sets could be an element of the non-relation part of class. (Contributed by RP, 25-Oct-2020.)

Assertion

Ref	Expression
elnonrel	|- ( <. X , Y >. e. ( A \ `' `' A ) <-> ( (/) e. A /\ -. ( X e. _V /\ Y e. _V ) ) )

Proof of Theorem elnonrel

Step	Hyp	Ref	Expression
1		nonrel 37890	. . 3 |- ( A \ `' `' A ) = ( A \ ( _V X. _V ) )
2	1	eleq2i 2693	. 2 |- ( <. X , Y >. e. ( A \ `' `' A ) <-> <. X , Y >. e. ( A \ ( _V X. _V ) ) )
3		eldif 3584	. . 3 |- ( <. X , Y >. e. ( A \ ( _V X. _V ) ) <-> ( <. X , Y >. e. A /\ -. <. X , Y >. e. ( _V X. _V ) ) )
4		opelxp 5146	. . . . . 6 |- ( <. X , Y >. e. ( _V X. _V ) <-> ( X e. _V /\ Y e. _V ) )
5	4	notbii 310	. . . . 5 |- ( -. <. X , Y >. e. ( _V X. _V ) <-> -. ( X e. _V /\ Y e. _V ) )
6	5	anbi2i 730	. . . 4 |- ( ( <. X , Y >. e. A /\ -. <. X , Y >. e. ( _V X. _V ) ) <-> ( <. X , Y >. e. A /\ -. ( X e. _V /\ Y e. _V ) ) )
7		opprc 4424	. . . . . 6 |- ( -. ( X e. _V /\ Y e. _V ) -> <. X , Y >. = (/) )
8	7	eleq1d 2686	. . . . 5 |- ( -. ( X e. _V /\ Y e. _V ) -> ( <. X , Y >. e. A <-> (/) e. A ) )
9	8	pm5.32ri 670	. . . 4 |- ( ( <. X , Y >. e. A /\ -. ( X e. _V /\ Y e. _V ) ) <-> ( (/) e. A /\ -. ( X e. _V /\ Y e. _V ) ) )
10	6, 9	bitri 264	. . 3 |- ( ( <. X , Y >. e. A /\ -. <. X , Y >. e. ( _V X. _V ) ) <-> ( (/) e. A /\ -. ( X e. _V /\ Y e. _V ) ) )
11	3, 10	bitri 264	. 2 |- ( <. X , Y >. e. ( A \ ( _V X. _V ) ) <-> ( (/) e. A /\ -. ( X e. _V /\ Y e. _V ) ) )
12	2, 11	bitri 264	1 |- ( <. X , Y >. e. ( A \ `' `' A ) <-> ( (/) e. A /\ -. ( X e. _V /\ Y e. _V ) ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   \ cdif 3571   (/)c0 3915   <.cop 4183   X. cxp 5112   `'ccnv 5113

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-eu 2474  df-mo 2475  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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by: (None)