Theorem enerOLD 8003

Description: Obsolete proof of ener 8002 as of 1-May-2021. Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
enerOLD	|- ~~ Er _V

Proof of Theorem enerOLD

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

Step	Hyp	Ref	Expression
1		relen 7960	. . . 4 |- Rel ~~
2	1	a1i 11	. . 3 |- ( T. -> Rel ~~ )
3		bren 7964	. . . . 5 |- ( x ~~ y <-> E. f f : x -1-1-onto-> y )
4		f1ocnv 6149	. . . . . . 7 |- ( f : x -1-1-onto-> y -> `' f : y -1-1-onto-> x )
5		vex 3203	. . . . . . . 8 |- y e. _V
6		vex 3203	. . . . . . . 8 |- x e. _V
7		f1oen2g 7972	. . . . . . . 8 |- ( ( y e. _V /\ x e. _V /\ `' f : y -1-1-onto-> x ) -> y ~~ x )
8	5, 6, 7	mp3an12 1414	. . . . . . 7 |- ( `' f : y -1-1-onto-> x -> y ~~ x )
9	4, 8	syl 17	. . . . . 6 |- ( f : x -1-1-onto-> y -> y ~~ x )
10	9	exlimiv 1858	. . . . 5 |- ( E. f f : x -1-1-onto-> y -> y ~~ x )
11	3, 10	sylbi 207	. . . 4 |- ( x ~~ y -> y ~~ x )
12	11	adantl 482	. . 3 |- ( ( T. /\ x ~~ y ) -> y ~~ x )
13		bren 7964	. . . . 5 |- ( x ~~ y <-> E. g g : x -1-1-onto-> y )
14		bren 7964	. . . . 5 |- ( y ~~ z <-> E. f f : y -1-1-onto-> z )
15		eeanv 2182	. . . . . 6 |- ( E. g E. f ( g : x -1-1-onto-> y /\ f : y -1-1-onto-> z ) <-> ( E. g g : x -1-1-onto-> y /\ E. f f : y -1-1-onto-> z ) )
16		f1oco 6159	. . . . . . . . 9 |- ( ( f : y -1-1-onto-> z /\ g : x -1-1-onto-> y ) -> ( f o. g ) : x -1-1-onto-> z )
17	16	ancoms 469	. . . . . . . 8 |- ( ( g : x -1-1-onto-> y /\ f : y -1-1-onto-> z ) -> ( f o. g ) : x -1-1-onto-> z )
18		vex 3203	. . . . . . . . 9 |- z e. _V
19		f1oen2g 7972	. . . . . . . . 9 |- ( ( x e. _V /\ z e. _V /\ ( f o. g ) : x -1-1-onto-> z ) -> x ~~ z )
20	6, 18, 19	mp3an12 1414	. . . . . . . 8 |- ( ( f o. g ) : x -1-1-onto-> z -> x ~~ z )
21	17, 20	syl 17	. . . . . . 7 |- ( ( g : x -1-1-onto-> y /\ f : y -1-1-onto-> z ) -> x ~~ z )
22	21	exlimivv 1860	. . . . . 6 |- ( E. g E. f ( g : x -1-1-onto-> y /\ f : y -1-1-onto-> z ) -> x ~~ z )
23	15, 22	sylbir 225	. . . . 5 |- ( ( E. g g : x -1-1-onto-> y /\ E. f f : y -1-1-onto-> z ) -> x ~~ z )
24	13, 14, 23	syl2anb 496	. . . 4 |- ( ( x ~~ y /\ y ~~ z ) -> x ~~ z )
25	24	adantl 482	. . 3 |- ( ( T. /\ ( x ~~ y /\ y ~~ z ) ) -> x ~~ z )
26	6	enref 7988	. . . . 5 |- x ~~ x
27	6, 26	2th 254	. . . 4 |- ( x e. _V <-> x ~~ x )
28	27	a1i 11	. . 3 |- ( T. -> ( x e. _V <-> x ~~ x ) )
29	2, 12, 25, 28	iserd 7768	. 2 |- ( T. -> ~~ Er _V )
30	29	trud 1493	1 |- ~~ Er _V

Syntax hints:   <-> wb 196   /\ wa 384   T. wtru 1484   E.wex 1704   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113   o. ccom 5118   Rel wrel 5119   -1-1-onto->wf1o 5887   Er wer 7739   ~~ cen 7952

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-er 7742  df-en 7956

This theorem is referenced by: (None)