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Theorem sbth 6997
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 6987 through sbthlem10 6996; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 6996. 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 6885 . . . 4  |-  Rel  ~<_
21brrelexi 4745 . . 3  |-  ( A  ~<_  B  ->  A  e.  _V )
31brrelexi 4745 . . 3  |-  ( B  ~<_  A  ->  B  e.  _V )
4 breq1 4042 . . . . . 6  |-  ( z  =  A  ->  (
z  ~<_  w  <->  A  ~<_  w ) )
5 breq2 4043 . . . . . 6  |-  ( z  =  A  ->  (
w  ~<_  z  <->  w  ~<_  A ) )
64, 5anbi12d 691 . . . . 5  |-  ( z  =  A  ->  (
( z  ~<_  w  /\  w  ~<_  z )  <->  ( A  ~<_  w  /\  w  ~<_  A ) ) )
7 breq1 4042 . . . . 5  |-  ( z  =  A  ->  (
z  ~~  w  <->  A  ~~  w ) )
86, 7imbi12d 311 . . . 4  |-  ( z  =  A  ->  (
( ( z  ~<_  w  /\  w  ~<_  z )  ->  z  ~~  w
)  <->  ( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w ) ) )
9 breq2 4043 . . . . . 6  |-  ( w  =  B  ->  ( A  ~<_  w  <->  A  ~<_  B ) )
10 breq1 4042 . . . . . 6  |-  ( w  =  B  ->  (
w  ~<_  A  <->  B  ~<_  A ) )
119, 10anbi12d 691 . . . . 5  |-  ( w  =  B  ->  (
( A  ~<_  w  /\  w  ~<_  A )  <->  ( A  ~<_  B  /\  B  ~<_  A ) ) )
12 breq2 4043 . . . . 5  |-  ( w  =  B  ->  ( A  ~~  w  <->  A  ~~  B ) )
1311, 12imbi12d 311 . . . 4  |-  ( w  =  B  ->  (
( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w
)  <->  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B ) ) )
14 vex 2804 . . . . 5  |-  z  e. 
_V
15 sseq1 3212 . . . . . . 7  |-  ( y  =  x  ->  (
y  C_  z  <->  x  C_  z
) )
16 imaeq2 5024 . . . . . . . . . 10  |-  ( y  =  x  ->  (
f " y )  =  ( f "
x ) )
1716difeq2d 3307 . . . . . . . . 9  |-  ( y  =  x  ->  (
w  \  ( f " y ) )  =  ( w  \ 
( f " x
) ) )
1817imaeq2d 5028 . . . . . . . 8  |-  ( y  =  x  ->  (
g " ( w 
\  ( f "
y ) ) )  =  ( g "
( w  \  (
f " x ) ) ) )
19 difeq2 3301 . . . . . . . 8  |-  ( y  =  x  ->  (
z  \  y )  =  ( z  \  x ) )
2018, 19sseq12d 3220 . . . . . . 7  |-  ( y  =  x  ->  (
( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y )  <->  ( g " ( w  \ 
( f " x
) ) )  C_  ( z  \  x
) ) )
2115, 20anbi12d 691 . . . . . 6  |-  ( y  =  x  ->  (
( y  C_  z  /\  ( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y ) )  <-> 
( x  C_  z  /\  ( g " (
w  \  ( f " x ) ) )  C_  ( z  \  x ) ) ) )
2221cbvabv 2415 . . . . 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 2296 . . . . 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 2804 . . . . 5  |-  w  e. 
_V
2514, 22, 23, 24sbthlem10 6996 . . . 4  |-  ( ( z  ~<_  w  /\  w  ~<_  z )  ->  z  ~~  w )
268, 13, 25vtocl2g 2860 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B
) )
272, 3, 26syl2an 463 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  (
( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B ) )
2827pm2.43i 43 1  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    \ cdif 3162    u. cun 3163    C_ wss 3165   U.cuni 3843   class class class wbr 4039   `'ccnv 4704    |` cres 4707   "cima 4708    ~~ cen 6876    ~<_ cdom 6877
This theorem is referenced by:  sbthb  6998  sdomnsym  7002  domtriord  7023  xpen  7040  limenpsi  7052  php  7061  onomeneq  7066  unbnn  7129  infxpenlem  7657  fseqen  7670  infpwfien  7705  inffien  7706  alephdom  7724  mappwen  7755  infcdaabs  7848  infunabs  7849  infcda  7850  infdif  7851  infxpabs  7854  infmap2  7860  gchhar  8309  gchaleph  8313  inttsk  8412  inar1  8413  xpnnenOLD  12504  znnen  12507  qnnen  12508  rpnnen  12521  rexpen  12522  mreexfidimd  13568  acsinfdimd  14301  fislw  14952  opnreen  18352  ovolctb2  18867  vitali  18984  aannenlem3  19726  basellem4  20337  lgsqrlem4  20599  umgraex  23890  sndw  25202  pellexlem4  27019  pellexlem5  27020  idomsubgmo  27616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-en 6880  df-dom 6881
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