Theorem cnv0OLD 5536

Description: Obsolete version of cnv0 5535 as of 25-Oct-2021. (Contributed by NM, 6-Apr-1998.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cnv0OLD	|- `' (/) = (/)

Proof of Theorem cnv0OLD

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

Step	Hyp	Ref	Expression
1		relcnv 5503	. 2 |- Rel `' (/)
2		rel0 5243	. 2 |- Rel (/)
3		vex 3203	. . . 4 |- x e. _V
4		vex 3203	. . . 4 |- y e. _V
5	3, 4	opelcnv 5304	. . 3 |- ( <. x , y >. e. `' (/) <-> <. y , x >. e. (/) )
6		noel 3919	. . . 4 |- -. <. x , y >. e. (/)
7		noel 3919	. . . 4 |- -. <. y , x >. e. (/)
8	6, 7	2false 365	. . 3 |- ( <. x , y >. e. (/) <-> <. y , x >. e. (/) )
9	5, 8	bitr4i 267	. 2 |- ( <. x , y >. e. `' (/) <-> <. x , y >. e. (/) )
10	1, 2, 9	eqrelriiv 5214	1 |- `' (/) = (/)

Syntax hints:   = wceq 1483   e. wcel 1990   (/)c0 3915   <.cop 4183   `'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-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)