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Theorem sbth 6976
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 6966 through sbthlem10 6975; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 6975. 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 )
Dummy variables  x  y  z  w  f 
g are mutually distinct and distinct from all other variables.

Proof of Theorem sbth
StepHypRef Expression
1 reldom 6864 . . . 4  |-  Rel  ~<_
21brrelexi 4728 . . 3  |-  ( A  ~<_  B  ->  A  e.  _V )
31brrelexi 4728 . . 3  |-  ( B  ~<_  A  ->  B  e.  _V )
4 breq1 4027 . . . . . 6  |-  ( z  =  A  ->  (
z  ~<_  w  <->  A  ~<_  w ) )
5 breq2 4028 . . . . . 6  |-  ( z  =  A  ->  (
w  ~<_  z  <->  w  ~<_  A ) )
64, 5anbi12d 693 . . . . 5  |-  ( z  =  A  ->  (
( z  ~<_  w  /\  w  ~<_  z )  <->  ( A  ~<_  w  /\  w  ~<_  A ) ) )
7 breq1 4027 . . . . 5  |-  ( z  =  A  ->  (
z  ~~  w  <->  A  ~~  w ) )
86, 7imbi12d 313 . . . 4  |-  ( z  =  A  ->  (
( ( z  ~<_  w  /\  w  ~<_  z )  ->  z  ~~  w
)  <->  ( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w ) ) )
9 breq2 4028 . . . . . 6  |-  ( w  =  B  ->  ( A  ~<_  w  <->  A  ~<_  B ) )
10 breq1 4027 . . . . . 6  |-  ( w  =  B  ->  (
w  ~<_  A  <->  B  ~<_  A ) )
119, 10anbi12d 693 . . . . 5  |-  ( w  =  B  ->  (
( A  ~<_  w  /\  w  ~<_  A )  <->  ( A  ~<_  B  /\  B  ~<_  A ) ) )
12 breq2 4028 . . . . 5  |-  ( w  =  B  ->  ( A  ~~  w  <->  A  ~~  B ) )
1311, 12imbi12d 313 . . . 4  |-  ( w  =  B  ->  (
( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w
)  <->  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B ) ) )
14 vex 2792 . . . . 5  |-  z  e. 
_V
15 sseq1 3200 . . . . . . 7  |-  ( y  =  x  ->  (
y  C_  z  <->  x  C_  z
) )
16 imaeq2 5007 . . . . . . . . . 10  |-  ( y  =  x  ->  (
f " y )  =  ( f "
x ) )
1716difeq2d 3295 . . . . . . . . 9  |-  ( y  =  x  ->  (
w  \  ( f " y ) )  =  ( w  \ 
( f " x
) ) )
1817imaeq2d 5011 . . . . . . . 8  |-  ( y  =  x  ->  (
g " ( w 
\  ( f "
y ) ) )  =  ( g "
( w  \  (
f " x ) ) ) )
19 difeq2 3289 . . . . . . . 8  |-  ( y  =  x  ->  (
z  \  y )  =  ( z  \  x ) )
2018, 19sseq12d 3208 . . . . . . 7  |-  ( y  =  x  ->  (
( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y )  <->  ( g " ( w  \ 
( f " x
) ) )  C_  ( z  \  x
) ) )
2115, 20anbi12d 693 . . . . . 6  |-  ( y  =  x  ->  (
( y  C_  z  /\  ( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y ) )  <-> 
( x  C_  z  /\  ( g " (
w  \  ( f " x ) ) )  C_  ( z  \  x ) ) ) )
2221cbvabv 2403 . . . . 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 2284 . . . . 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 2792 . . . . 5  |-  w  e. 
_V
2514, 22, 23, 24sbthlem10 6975 . . . 4  |-  ( ( z  ~<_  w  /\  w  ~<_  z )  ->  z  ~~  w )
268, 13, 25vtocl2g 2848 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B
) )
272, 3, 26syl2an 465 . 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 6    /\ wa 360    = wceq 1624    e. wcel 1685   {cab 2270   _Vcvv 2789    \ cdif 3150    u. cun 3151    C_ wss 3153   U.cuni 3828   class class class wbr 4024   `'ccnv 4687    |` cres 4690   "cima 4691    ~~ cen 6855    ~<_ cdom 6856
This theorem is referenced by:  sbthb  6977  sdomnsym  6981  domtriord  7002  xpen  7019  limenpsi  7031  php  7040  onomeneq  7045  unbnn  7108  infxpenlem  7636  fseqen  7649  infpwfien  7684  inffien  7685  alephdom  7703  mappwen  7734  infcdaabs  7827  infunabs  7828  infcda  7829  infdif  7830  infxpabs  7833  infmap2  7839  gchhar  8288  gchaleph  8292  inttsk  8391  inar1  8392  xpnnenOLD  12482  znnen  12485  qnnen  12486  rpnnen  12499  rexpen  12500  mreexfidimd  13546  acsinfdimd  14279  fislw  14930  opnreen  18330  ovolctb2  18845  vitali  18962  aannenlem3  19704  basellem4  20315  lgsqrlem4  20577  umgraex  23279  sndw  24498  pellexlem4  26316  pellexlem5  26317  idomsubgmo  26913
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-en 6859  df-dom 6860
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