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Theorem sbthb 7639
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
StepHypRef Expression
1 sbth 7638 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B )
2 endom 7543 . . 3  |-  ( A 
~~  B  ->  A  ~<_  B )
3 ensym 7565 . . . 4  |-  ( A 
~~  B  ->  B  ~~  A )
4 endom 7543 . . . 4  |-  ( B 
~~  A  ->  B  ~<_  A )
53, 4syl 16 . . 3  |-  ( A 
~~  B  ->  B  ~<_  A )
62, 5jca 532 . 2  |-  ( A 
~~  B  ->  ( A  ~<_  B  /\  B  ~<_  A ) )
71, 6impbii 188 1  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  <->  A  ~~  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   class class class wbr 4447    ~~ cen 7514    ~<_ cdom 7515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-er 7312  df-en 7518  df-dom 7519
This theorem is referenced by:  sbthcl  7640  dom0  7646  carden2  8369  axgroth2  9204
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