Theorem 0erOLD 7781

Description: Obsolete proof of 0er 7780 as of 1-May-2021. The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
0erOLD	|- (/) Er (/)

Proof of Theorem 0erOLD

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

Step	Hyp	Ref	Expression
1		rel0 5243	. . . 4 |- Rel (/)
2	1	a1i 11	. . 3 |- ( T. -> Rel (/) )
3		df-br 4654	. . . . 5 |- ( x (/) y <-> <. x , y >. e. (/) )
4		noel 3919	. . . . . 6 |- -. <. x , y >. e. (/)
5	4	pm2.21i 116	. . . . 5 |- ( <. x , y >. e. (/) -> y (/) x )
6	3, 5	sylbi 207	. . . 4 |- ( x (/) y -> y (/) x )
7	6	adantl 482	. . 3 |- ( ( T. /\ x (/) y ) -> y (/) x )
8	4	pm2.21i 116	. . . . 5 |- ( <. x , y >. e. (/) -> x (/) z )
9	3, 8	sylbi 207	. . . 4 |- ( x (/) y -> x (/) z )
10	9	ad2antrl 764	. . 3 |- ( ( T. /\ ( x (/) y /\ y (/) z ) ) -> x (/) z )
11		noel 3919	. . . . . 6 |- -. x e. (/)
12		noel 3919	. . . . . 6 |- -. <. x , x >. e. (/)
13	11, 12	2false 365	. . . . 5 |- ( x e. (/) <-> <. x , x >. e. (/) )
14		df-br 4654	. . . . 5 |- ( x (/) x <-> <. x , x >. e. (/) )
15	13, 14	bitr4i 267	. . . 4 |- ( x e. (/) <-> x (/) x )
16	15	a1i 11	. . 3 |- ( T. -> ( x e. (/) <-> x (/) x ) )
17	2, 7, 10, 16	iserd 7768	. 2 |- ( T. -> (/) Er (/) )
18	17	trud 1493	1 |- (/) Er (/)

Syntax hints:   <-> wb 196   T. wtru 1484   e. wcel 1990   (/)c0 3915   <.cop 4183   class class class wbr 4653   Rel wrel 5119   Er wer 7739

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  df-co 5123  df-dm 5124  df-er 7742

This theorem is referenced by: (None)