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Theorem sbth 7218
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 7208 through sbthlem10 7217; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 7217. 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 7106 . . . 4  |-  Rel  ~<_
21brrelexi 4909 . . 3  |-  ( A  ~<_  B  ->  A  e.  _V )
31brrelexi 4909 . . 3  |-  ( B  ~<_  A  ->  B  e.  _V )
4 breq1 4207 . . . . . 6  |-  ( z  =  A  ->  (
z  ~<_  w  <->  A  ~<_  w ) )
5 breq2 4208 . . . . . 6  |-  ( z  =  A  ->  (
w  ~<_  z  <->  w  ~<_  A ) )
64, 5anbi12d 692 . . . . 5  |-  ( z  =  A  ->  (
( z  ~<_  w  /\  w  ~<_  z )  <->  ( A  ~<_  w  /\  w  ~<_  A ) ) )
7 breq1 4207 . . . . 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 4208 . . . . . 6  |-  ( w  =  B  ->  ( A  ~<_  w  <->  A  ~<_  B ) )
10 breq1 4207 . . . . . 6  |-  ( w  =  B  ->  (
w  ~<_  A  <->  B  ~<_  A ) )
119, 10anbi12d 692 . . . . 5  |-  ( w  =  B  ->  (
( A  ~<_  w  /\  w  ~<_  A )  <->  ( A  ~<_  B  /\  B  ~<_  A ) ) )
12 breq2 4208 . . . . 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 2951 . . . . 5  |-  z  e. 
_V
15 sseq1 3361 . . . . . . 7  |-  ( y  =  x  ->  (
y  C_  z  <->  x  C_  z
) )
16 imaeq2 5190 . . . . . . . . . 10  |-  ( y  =  x  ->  (
f " y )  =  ( f "
x ) )
1716difeq2d 3457 . . . . . . . . 9  |-  ( y  =  x  ->  (
w  \  ( f " y ) )  =  ( w  \ 
( f " x
) ) )
1817imaeq2d 5194 . . . . . . . 8  |-  ( y  =  x  ->  (
g " ( w 
\  ( f "
y ) ) )  =  ( g "
( w  \  (
f " x ) ) ) )
19 difeq2 3451 . . . . . . . 8  |-  ( y  =  x  ->  (
z  \  y )  =  ( z  \  x ) )
2018, 19sseq12d 3369 . . . . . . 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 2554 . . . . 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 2435 . . . . 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 2951 . . . . 5  |-  w  e. 
_V
2514, 22, 23, 24sbthlem10 7217 . . . 4  |-  ( ( z  ~<_  w  /\  w  ~<_  z )  ->  z  ~~  w )
268, 13, 25vtocl2g 3007 . . 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 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    \ cdif 3309    u. cun 3310    C_ wss 3312   U.cuni 4007   class class class wbr 4204   `'ccnv 4868    |` cres 4871   "cima 4872    ~~ cen 7097    ~<_ cdom 7098
This theorem is referenced by:  sbthb  7219  sdomnsym  7223  domtriord  7244  xpen  7261  limenpsi  7273  php  7282  onomeneq  7287  unbnn  7354  infxpenlem  7884  fseqen  7897  infpwfien  7932  inffien  7933  alephdom  7951  mappwen  7982  infcdaabs  8075  infunabs  8076  infcda  8077  infdif  8078  infxpabs  8081  infmap2  8087  gchhar  8535  gchaleph  8539  inttsk  8638  inar1  8639  xpnnenOLD  12797  znnen  12800  qnnen  12801  rpnnen  12814  rexpen  12815  mreexfidimd  13863  acsinfdimd  14596  fislw  15247  opnreen  18850  ovolctb2  19376  vitali  19493  aannenlem3  20235  basellem4  20854  lgsqrlem4  21116  umgraex  21346  pellexlem4  26832  pellexlem5  26833  idomsubgmo  27429
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-en 7101  df-dom 7102
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