Theorem sbthcl 8082

Description: Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.)

Assertion

Ref	Expression
sbthcl	|- ~~ = ( ~<_ i^i `' ~<_ )

Proof of Theorem sbthcl

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

Step	Hyp	Ref	Expression
1		relen 7960	. 2 |- Rel ~~
2		inss1 3833	. . 3 |- ( ~<_ i^i `' ~<_ ) C_ ~<_
3		reldom 7961	. . 3 |- Rel ~<_
4		relss 5206	. . 3 |- ( ( ~<_ i^i `' ~<_ ) C_ ~<_ -> ( Rel ~<_ -> Rel ( ~<_ i^i `' ~<_ ) ) )
5	2, 3, 4	mp2 9	. 2 |- Rel ( ~<_ i^i `' ~<_ )
6		brin 4704	. . 3 |- ( x ( ~<_ i^i `' ~<_ ) y <-> ( x ~<_ y /\ x `' ~<_ y ) )
7		vex 3203	. . . . 5 |- x e. _V
8		vex 3203	. . . . 5 |- y e. _V
9	7, 8	brcnv 5305	. . . 4 |- ( x `' ~<_ y <-> y ~<_ x )
10	9	anbi2i 730	. . 3 |- ( ( x ~<_ y /\ x `' ~<_ y ) <-> ( x ~<_ y /\ y ~<_ x ) )
11		sbthb 8081	. . 3 |- ( ( x ~<_ y /\ y ~<_ x ) <-> x ~~ y )
12	6, 10, 11	3bitrri 287	. 2 |- ( x ~~ y <-> x ( ~<_ i^i `' ~<_ ) y )
13	1, 5, 12	eqbrriv 5215	1 |- ~~ = ( ~<_ i^i `' ~<_ )

Syntax hints:   /\ wa 384   = wceq 1483   i^i cin 3573   C_ wss 3574   class class class wbr 4653   `'ccnv 5113   Rel wrel 5119   ~~ cen 7952   ~<_ cdom 7953

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  df-dom 7957

This theorem is referenced by:  dfsdom2  8083