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Theorem exmidsbth 15519
Description: The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 7026) 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 7026.

The reverse direction (exmidsbthr 15518) 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 7026 . . . 4  |-  ( (EXMID  /\  ( x  ~<_  y  /\  y  ~<_  x ) )  ->  x  ~~  y
)
21ex 115 . . 3  |-  (EXMID  ->  (
( x  ~<_  y  /\  y  ~<_  x )  ->  x  ~~  y ) )
32alrimivv 1886 . 2  |-  (EXMID  ->  A. x A. y ( ( x  ~<_  y  /\  y  ~<_  x )  ->  x  ~~  y ) )
4 exmidsbthr 15518 . 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 1362   class class class wbr 4029  EXMIDwem 4223    ~~ cen 6792    ~<_ cdom 6793
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-exmid 4224  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-1o 6469  df-2o 6470  df-map 6704  df-en 6795  df-dom 6796  df-dju 7097  df-inl 7106  df-inr 7107  df-case 7143  df-nninf 7179  df-omni 7194
This theorem is referenced by: (None)
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