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Theorem sbthb 7228
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 7227 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B )
2 endom 7134 . . 3  |-  ( A 
~~  B  ->  A  ~<_  B )
3 ensym 7156 . . . 4  |-  ( A 
~~  B  ->  B  ~~  A )
4 endom 7134 . . . 4  |-  ( B 
~~  A  ->  B  ~<_  A )
53, 4syl 16 . . 3  |-  ( A 
~~  B  ->  B  ~<_  A )
62, 5jca 519 . 2  |-  ( A 
~~  B  ->  ( A  ~<_  B  /\  B  ~<_  A ) )
71, 6impbii 181 1  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  <->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   class class class wbr 4212    ~~ cen 7106    ~<_ cdom 7107
This theorem is referenced by:  sbthcl  7229  dom0  7235  carden2  7874  axgroth2  8700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-er 6905  df-en 7110  df-dom 7111
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