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Theorem sbth 7186
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 7176 through sbthlem10 7185; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 7185. 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. (Contributed by NM, 8-Jun-1998.)
Assertion
Ref Expression
sbth  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B )

Proof of Theorem sbth
Dummy variables  x  y  z  w  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldom 7074 . . . 4  |-  Rel  ~<_
21brrelexi 4877 . . 3  |-  ( A  ~<_  B  ->  A  e.  _V )
31brrelexi 4877 . . 3  |-  ( B  ~<_  A  ->  B  e.  _V )
4 breq1 4175 . . . . . 6  |-  ( z  =  A  ->  (
z  ~<_  w  <->  A  ~<_  w ) )
5 breq2 4176 . . . . . 6  |-  ( z  =  A  ->  (
w  ~<_  z  <->  w  ~<_  A ) )
64, 5anbi12d 692 . . . . 5  |-  ( z  =  A  ->  (
( z  ~<_  w  /\  w  ~<_  z )  <->  ( A  ~<_  w  /\  w  ~<_  A ) ) )
7 breq1 4175 . . . . 5  |-  ( z  =  A  ->  (
z  ~~  w  <->  A  ~~  w ) )
86, 7imbi12d 312 . . . 4  |-  ( z  =  A  ->  (
( ( z  ~<_  w  /\  w  ~<_  z )  ->  z  ~~  w
)  <->  ( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w ) ) )
9 breq2 4176 . . . . . 6  |-  ( w  =  B  ->  ( A  ~<_  w  <->  A  ~<_  B ) )
10 breq1 4175 . . . . . 6  |-  ( w  =  B  ->  (
w  ~<_  A  <->  B  ~<_  A ) )
119, 10anbi12d 692 . . . . 5  |-  ( w  =  B  ->  (
( A  ~<_  w  /\  w  ~<_  A )  <->  ( A  ~<_  B  /\  B  ~<_  A ) ) )
12 breq2 4176 . . . . 5  |-  ( w  =  B  ->  ( A  ~~  w  <->  A  ~~  B ) )
1311, 12imbi12d 312 . . . 4  |-  ( w  =  B  ->  (
( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w
)  <->  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B ) ) )
14 vex 2919 . . . . 5  |-  z  e. 
_V
15 sseq1 3329 . . . . . . 7  |-  ( y  =  x  ->  (
y  C_  z  <->  x  C_  z
) )
16 imaeq2 5158 . . . . . . . . . 10  |-  ( y  =  x  ->  (
f " y )  =  ( f "
x ) )
1716difeq2d 3425 . . . . . . . . 9  |-  ( y  =  x  ->  (
w  \  ( f " y ) )  =  ( w  \ 
( f " x
) ) )
1817imaeq2d 5162 . . . . . . . 8  |-  ( y  =  x  ->  (
g " ( w 
\  ( f "
y ) ) )  =  ( g "
( w  \  (
f " x ) ) ) )
19 difeq2 3419 . . . . . . . 8  |-  ( y  =  x  ->  (
z  \  y )  =  ( z  \  x ) )
2018, 19sseq12d 3337 . . . . . . 7  |-  ( y  =  x  ->  (
( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y )  <->  ( g " ( w  \ 
( f " x
) ) )  C_  ( z  \  x
) ) )
2115, 20anbi12d 692 . . . . . 6  |-  ( y  =  x  ->  (
( y  C_  z  /\  ( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y ) )  <-> 
( x  C_  z  /\  ( g " (
w  \  ( f " x ) ) )  C_  ( z  \  x ) ) ) )
2221cbvabv 2523 . . . . 5  |-  { y  |  ( y  C_  z  /\  ( g "
( w  \  (
f " y ) ) )  C_  (
z  \  y )
) }  =  {
x  |  ( x 
C_  z  /\  (
g " ( w 
\  ( f "
x ) ) ) 
C_  ( z  \  x ) ) }
23 eqid 2404 . . . . 5  |-  ( ( 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 ) ) } ) ) )
24 vex 2919 . . . . 5  |-  w  e. 
_V
2514, 22, 23, 24sbthlem10 7185 . . . 4  |-  ( ( z  ~<_  w  /\  w  ~<_  z )  ->  z  ~~  w )
268, 13, 25vtocl2g 2975 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B
) )
272, 3, 26syl2an 464 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  (
( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B ) )
2827pm2.43i 45 1  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   _Vcvv 2916    \ cdif 3277    u. cun 3278    C_ wss 3280   U.cuni 3975   class class class wbr 4172   `'ccnv 4836    |` cres 4839   "cima 4840    ~~ cen 7065    ~<_ cdom 7066
This theorem is referenced by:  sbthb  7187  sdomnsym  7191  domtriord  7212  xpen  7229  limenpsi  7241  php  7250  onomeneq  7255  unbnn  7322  infxpenlem  7851  fseqen  7864  infpwfien  7899  inffien  7900  alephdom  7918  mappwen  7949  infcdaabs  8042  infunabs  8043  infcda  8044  infdif  8045  infxpabs  8048  infmap2  8054  gchhar  8502  gchaleph  8506  inttsk  8605  inar1  8606  xpnnenOLD  12764  znnen  12767  qnnen  12768  rpnnen  12781  rexpen  12782  mreexfidimd  13830  acsinfdimd  14563  fislw  15214  opnreen  18815  ovolctb2  19341  vitali  19458  aannenlem3  20200  basellem4  20819  lgsqrlem4  21081  umgraex  21311  pellexlem4  26785  pellexlem5  26786  idomsubgmo  27382
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-en 7069  df-dom 7070
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