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Theorem exmidsbth 14775
Description: The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 6966) is the proof of the Schroeder-Bernstein Theorem from the Metamath Proof Explorer database (in which excluded middle holds), but adapted to use EXMID as an antecedent rather than being unconditionally true, as in the non-intuitionistic proof at https://us.metamath.org/mpeuni/sbth.html 6966.

The reverse direction (exmidsbthr 14774) is the one which establishes that Schroeder-Bernstein implies excluded middle. This resolves the question of whether we will be able to prove Schroeder-Bernstein from our axioms in the negative. (Contributed by Jim Kingdon, 13-Aug-2022.)

Assertion
Ref Expression
exmidsbth  |-  (EXMID  <->  A. x A. y ( ( x  ~<_  y  /\  y  ~<_  x )  ->  x  ~~  y ) )
Distinct variable group:    x, y

Proof of Theorem exmidsbth
StepHypRef Expression
1 isbth 6966 . . . 4  |-  ( (EXMID  /\  ( x  ~<_  y  /\  y  ~<_  x ) )  ->  x  ~~  y
)
21ex 115 . . 3  |-  (EXMID  ->  (
( x  ~<_  y  /\  y  ~<_  x )  ->  x  ~~  y ) )
32alrimivv 1875 . 2  |-  (EXMID  ->  A. x A. y ( ( x  ~<_  y  /\  y  ~<_  x )  ->  x  ~~  y ) )
4 exmidsbthr 14774 . 2  |-  ( A. x A. y ( ( x  ~<_  y  /\  y  ~<_  x )  ->  x  ~~  y )  -> EXMID )
53, 4impbii 126 1  |-  (EXMID  <->  A. x A. y ( ( x  ~<_  y  /\  y  ~<_  x )  ->  x  ~~  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1351   class class class wbr 4004  EXMIDwem 4195    ~~ cen 6738    ~<_ cdom 6739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-exmid 4196  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-1o 6417  df-2o 6418  df-map 6650  df-en 6741  df-dom 6742  df-dju 7037  df-inl 7046  df-inr 7047  df-case 7083  df-nninf 7119  df-omni 7133
This theorem is referenced by: (None)
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