Theorem n0el2 34103

Description: Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 31-Jan-2018.)

Assertion

Ref	Expression
n0el2	|- ( -. (/) e. A <-> dom ( `' _E |` A ) = A )

Proof of Theorem n0el2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmopab3 5337	. 2 |- ( A. x e. A E. y y e. x <-> dom { <. x , y >. | ( x e. A /\ y e. x ) } = A )
2		n0el 3940	. 2 |- ( -. (/) e. A <-> A. x e. A E. y y e. x )
3		cnvepres 34066	. . . 4 |- ( `' _E |` A ) = { <. x , y >. | ( x e. A /\ y e. x ) }
4	3	dmeqi 5325	. . 3 |- dom ( `' _E |` A ) = dom { <. x , y >. | ( x e. A /\ y e. x ) }
5	4	eqeq1i 2627	. 2 |- ( dom ( `' _E |` A ) = A <-> dom { <. x , y >. | ( x e. A /\ y e. x ) } = A )
6	1, 2, 5	3bitr4i 292	1 |- ( -. (/) e. A <-> dom ( `' _E |` A ) = A )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   (/)c0 3915   {copab 4712   _E cep 5028   `'ccnv 5113   dom cdm 5114   |` cres 5116

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-ne 2795  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-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-res 5126

This theorem is referenced by: (None)