Theorem sbthb 8081

Description: Schroeder-Bernstein Theorem and its converse. (Contributed by NM, 8-Jun-1998.)

Assertion

Ref	Expression
sbthb	|- ( ( A ~<_ B /\ B ~<_ A ) <-> A ~~ B )

Proof of Theorem sbthb

Step	Hyp	Ref	Expression
1		sbth 8080	. 2 |- ( ( A ~<_ B /\ B ~<_ A ) -> A ~~ B )
2		endom 7982	. . 3 |- ( A ~~ B -> A ~<_ B )
3		ensym 8005	. . . 4 |- ( A ~~ B -> B ~~ A )
4		endom 7982	. . . 4 |- ( B ~~ A -> B ~<_ A )
5	3, 4	syl 17	. . 3 |- ( A ~~ B -> B ~<_ A )
6	2, 5	jca 554	. 2 |- ( A ~~ B -> ( A ~<_ B /\ B ~<_ A ) )
7	1, 6	impbii 199	1 |- ( ( A ~<_ B /\ B ~<_ A ) <-> A ~~ B )

Syntax hints:   <-> wb 196   /\ wa 384   class class class wbr 4653   ~~ 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:  sbthcl  8082  dom0  8088  carden2  8813  axgroth2  9647