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Theorem sbth 7429
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 7419 through sbthlem10 7428; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 7428. 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. This is Metamath 100 proof #25. (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 7314 . . . 4  |-  Rel  ~<_
21brrelexi 4877 . . 3  |-  ( A  ~<_  B  ->  A  e.  _V )
31brrelexi 4877 . . 3  |-  ( B  ~<_  A  ->  B  e.  _V )
4 breq1 4293 . . . . . 6  |-  ( z  =  A  ->  (
z  ~<_  w  <->  A  ~<_  w ) )
5 breq2 4294 . . . . . 6  |-  ( z  =  A  ->  (
w  ~<_  z  <->  w  ~<_  A ) )
64, 5anbi12d 710 . . . . 5  |-  ( z  =  A  ->  (
( z  ~<_  w  /\  w  ~<_  z )  <->  ( A  ~<_  w  /\  w  ~<_  A ) ) )
7 breq1 4293 . . . . 5  |-  ( z  =  A  ->  (
z  ~~  w  <->  A  ~~  w ) )
86, 7imbi12d 320 . . . 4  |-  ( z  =  A  ->  (
( ( z  ~<_  w  /\  w  ~<_  z )  ->  z  ~~  w
)  <->  ( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w ) ) )
9 breq2 4294 . . . . . 6  |-  ( w  =  B  ->  ( A  ~<_  w  <->  A  ~<_  B ) )
10 breq1 4293 . . . . . 6  |-  ( w  =  B  ->  (
w  ~<_  A  <->  B  ~<_  A ) )
119, 10anbi12d 710 . . . . 5  |-  ( w  =  B  ->  (
( A  ~<_  w  /\  w  ~<_  A )  <->  ( A  ~<_  B  /\  B  ~<_  A ) ) )
12 breq2 4294 . . . . 5  |-  ( w  =  B  ->  ( A  ~~  w  <->  A  ~~  B ) )
1311, 12imbi12d 320 . . . 4  |-  ( w  =  B  ->  (
( ( A  ~<_  w  /\  w  ~<_  A )  ->  A  ~~  w
)  <->  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B ) ) )
14 vex 2973 . . . . 5  |-  z  e. 
_V
15 sseq1 3375 . . . . . . 7  |-  ( y  =  x  ->  (
y  C_  z  <->  x  C_  z
) )
16 imaeq2 5163 . . . . . . . . . 10  |-  ( y  =  x  ->  (
f " y )  =  ( f "
x ) )
1716difeq2d 3472 . . . . . . . . 9  |-  ( y  =  x  ->  (
w  \  ( f " y ) )  =  ( w  \ 
( f " x
) ) )
1817imaeq2d 5167 . . . . . . . 8  |-  ( y  =  x  ->  (
g " ( w 
\  ( f "
y ) ) )  =  ( g "
( w  \  (
f " x ) ) ) )
19 difeq2 3466 . . . . . . . 8  |-  ( y  =  x  ->  (
z  \  y )  =  ( z  \  x ) )
2018, 19sseq12d 3383 . . . . . . 7  |-  ( y  =  x  ->  (
( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y )  <->  ( g " ( w  \ 
( f " x
) ) )  C_  ( z  \  x
) ) )
2115, 20anbi12d 710 . . . . . 6  |-  ( y  =  x  ->  (
( y  C_  z  /\  ( g " (
w  \  ( f " y ) ) )  C_  ( z  \  y ) )  <-> 
( x  C_  z  /\  ( g " (
w  \  ( f " x ) ) )  C_  ( z  \  x ) ) ) )
2221cbvabv 2560 . . . . 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 2441 . . . . 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 2973 . . . . 5  |-  w  e. 
_V
2514, 22, 23, 24sbthlem10 7428 . . . 4  |-  ( ( z  ~<_  w  /\  w  ~<_  z )  ->  z  ~~  w )
268, 13, 25vtocl2g 3032 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B
) )
272, 3, 26syl2an 477 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  (
( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B ) )
2827pm2.43i 47 1  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2427   _Vcvv 2970    \ cdif 3323    u. cun 3324    C_ wss 3326   U.cuni 4089   class class class wbr 4290   `'ccnv 4837    |` cres 4840   "cima 4841    ~~ cen 7305    ~<_ cdom 7306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-en 7309  df-dom 7310
This theorem is referenced by:  sbthb  7430  sdomnsym  7434  domtriord  7455  xpen  7472  limenpsi  7484  php  7493  onomeneq  7498  unbnn  7566  infxpenlem  8178  fseqen  8195  infpwfien  8230  inffien  8231  alephdom  8249  mappwen  8280  infcdaabs  8373  infunabs  8374  infcda  8375  infdif  8376  infxpabs  8379  infmap2  8385  gchaleph  8836  gchhar  8844  inttsk  8939  inar1  8940  xpnnenOLD  13490  znnen  13493  qnnen  13494  rpnnen  13507  rexpen  13508  mreexfidimd  14586  acsinfdimd  15350  fislw  16122  opnreen  20406  ovolctb2  20973  vitali  21091  aannenlem3  21794  basellem4  22419  lgsqrlem4  22681  umgraex  23255  pellexlem4  29170  pellexlem5  29171  idomsubgmo  29560
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