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Theorem isbth 6821
Description: Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set 
A is smaller (has lower cardinality) than  B and vice-versa, then  A and  B are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 6811 through sbthlemi10 6820; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 6820. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. The proof does require the law of the excluded middle which cannot be avoided as shown at exmidsbthr 13029. (Contributed by NM, 8-Jun-1998.)
Assertion
Ref Expression
isbth  |-  ( (EXMID  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  A  ~~  B
)

Proof of Theorem isbth
Dummy variables  x  y  z  w  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 503 . 2  |-  ( (EXMID  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  A  ~<_  B )
2 simprr 504 . 2  |-  ( (EXMID  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  B  ~<_  A )
3 reldom 6605 . . . . 5  |-  Rel  ~<_
43brrelex1i 4550 . . . 4  |-  ( B  ~<_  A  ->  B  e.  _V )
52, 4syl 14 . . 3  |-  ( (EXMID  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  B  e.  _V )
6 breq2 3901 . . . . . 6  |-  ( w  =  B  ->  ( A  ~<_  w  <->  A  ~<_  B ) )
7 breq1 3900 . . . . . 6  |-  ( w  =  B  ->  (
w  ~<_  A  <->  B  ~<_  A ) )
86, 7anbi12d 462 . . . . 5  |-  ( w  =  B  ->  (
( A  ~<_  w  /\  w  ~<_  A )  <->  ( A  ~<_  B  /\  B  ~<_  A ) ) )
9 breq2 3901 . . . . 5  |-  ( w  =  B  ->  ( A  ~~  w  <->  A  ~~  B ) )
108, 9imbi12d 233 . . . 4  |-  ( w  =  B  ->  (
( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w
)  <->  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B ) ) )
1110adantl 273 . . 3  |-  ( ( (EXMID 
/\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  w  =  B )  ->  (
( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w
)  <->  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B ) ) )
123brrelex1i 4550 . . . . 5  |-  ( A  ~<_  B  ->  A  e.  _V )
131, 12syl 14 . . . 4  |-  ( (EXMID  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  A  e.  _V )
14 breq1 3900 . . . . . . 7  |-  ( z  =  A  ->  (
z  ~<_  w  <->  A  ~<_  w ) )
15 breq2 3901 . . . . . . 7  |-  ( z  =  A  ->  (
w  ~<_  z  <->  w  ~<_  A ) )
1614, 15anbi12d 462 . . . . . 6  |-  ( z  =  A  ->  (
( z  ~<_  w  /\  w  ~<_  z )  <->  ( A  ~<_  w  /\  w  ~<_  A ) ) )
17 breq1 3900 . . . . . 6  |-  ( z  =  A  ->  (
z  ~~  w  <->  A  ~~  w ) )
1816, 17imbi12d 233 . . . . 5  |-  ( z  =  A  ->  (
( ( z  ~<_  w  /\  w  ~<_  z )  ->  z  ~~  w
)  <->  ( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w ) ) )
1918adantl 273 . . . 4  |-  ( ( (EXMID 
/\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  z  =  A )  ->  (
( ( z  ~<_  w  /\  w  ~<_  z )  ->  z  ~~  w
)  <->  ( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w ) ) )
20 vex 2661 . . . . . . 7  |-  z  e. 
_V
21 sseq1 3088 . . . . . . . . 9  |-  ( y  =  x  ->  (
y  C_  z  <->  x  C_  z
) )
22 imaeq2 4845 . . . . . . . . . . . 12  |-  ( y  =  x  ->  (
f " y )  =  ( f "
x ) )
2322difeq2d 3162 . . . . . . . . . . 11  |-  ( y  =  x  ->  (
w  \  ( f " y ) )  =  ( w  \ 
( f " x
) ) )
2423imaeq2d 4849 . . . . . . . . . 10  |-  ( y  =  x  ->  (
g " ( w 
\  ( f "
y ) ) )  =  ( g "
( w  \  (
f " x ) ) ) )
25 difeq2 3156 . . . . . . . . . 10  |-  ( y  =  x  ->  (
z  \  y )  =  ( z  \  x ) )
2624, 25sseq12d 3096 . . . . . . . . 9  |-  ( y  =  x  ->  (
( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y )  <->  ( g " ( w  \ 
( f " x
) ) )  C_  ( z  \  x
) ) )
2721, 26anbi12d 462 . . . . . . . 8  |-  ( y  =  x  ->  (
( y  C_  z  /\  ( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y ) )  <-> 
( x  C_  z  /\  ( g " (
w  \  ( f " x ) ) )  C_  ( z  \  x ) ) ) )
2827cbvabv 2239 . . . . . . 7  |-  { y  |  ( y  C_  z  /\  ( g "
( w  \  (
f " y ) ) )  C_  (
z  \  y )
) }  =  {
x  |  ( x 
C_  z  /\  (
g " ( w 
\  ( f "
x ) ) ) 
C_  ( z  \  x ) ) }
29 eqid 2115 . . . . . . 7  |-  ( ( f  |`  U. { y  |  ( y  C_  z  /\  ( g "
( w  \  (
f " y ) ) )  C_  (
z  \  y )
) } )  u.  ( `' g  |`  ( z  \  U. { y  |  ( y  C_  z  /\  ( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y ) ) } ) ) )  =  ( ( f  |`  U. { y  |  ( y  C_  z  /\  ( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y ) ) } )  u.  ( `' g  |`  ( z 
\  U. { y  |  ( y  C_  z  /\  ( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y ) ) } ) ) )
30 vex 2661 . . . . . . 7  |-  w  e. 
_V
3120, 28, 29, 30sbthlemi10 6820 . . . . . 6  |-  ( (EXMID  /\  ( z  ~<_  w  /\  w  ~<_  z ) )  ->  z  ~~  w
)
3231ex 114 . . . . 5  |-  (EXMID  ->  (
( z  ~<_  w  /\  w  ~<_  z )  -> 
z  ~~  w )
)
3332adantr 272 . . . 4  |-  ( (EXMID  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  ( ( z  ~<_  w  /\  w  ~<_  z )  ->  z  ~~  w ) )
3413, 19, 33vtocld 2710 . . 3  |-  ( (EXMID  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  ( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w ) )
355, 11, 34vtocld 2710 . 2  |-  ( (EXMID  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B ) )
361, 2, 35mp2and 427 1  |-  ( (EXMID  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  A  ~~  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   {cab 2101   _Vcvv 2658    \ cdif 3036    u. cun 3037    C_ wss 3039   U.cuni 3704   class class class wbr 3897  EXMIDwem 4086   `'ccnv 4506    |` cres 4509   "cima 4510    ~~ cen 6598    ~<_ cdom 6599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-exmid 4087  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-en 6601  df-dom 6602
This theorem is referenced by:  exmidsbth  13030
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