Theorem nonrel 37890

Description: A non-relation is equal to the base class with all ordered pairs removed. (Contributed by RP, 25-Oct-2020.)

Assertion

Ref	Expression
nonrel	|- ( A \ `' `' A ) = ( A \ ( _V X. _V ) )

Proof of Theorem nonrel

Step	Hyp	Ref	Expression
1		cnvcnv 5586	. . 3 |- `' `' A = ( A i^i ( _V X. _V ) )
2	1	difeq2i 3725	. 2 |- ( A \ `' `' A ) = ( A \ ( A i^i ( _V X. _V ) ) )
3		difin 3861	. 2 |- ( A \ ( A i^i ( _V X. _V ) ) ) = ( A \ ( _V X. _V ) )
4	2, 3	eqtri 2644	1 |- ( A \ `' `' A ) = ( A \ ( _V X. _V ) )

Syntax hints:   = wceq 1483   _Vcvv 3200   \ cdif 3571   i^i cin 3573   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-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:  elnonrel  37891