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Theorem exmidsbth 11869
Description: The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The Metamath Proof Explorer (MPE) database uses classical logic, which accepts excluded middle as an axiom. Thus the Schroeder-Bernstein Theorem is true in MPE; see https://us.metamath.org/mpeuni/sbth.html. (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 6674 . . . 4  |-  ( (EXMID  /\  ( x  ~<_  y  /\  y  ~<_  x ) )  ->  x  ~~  y
)
21ex 113 . . 3  |-  (EXMID  ->  (
( x  ~<_  y  /\  y  ~<_  x )  ->  x  ~~  y ) )
32alrimivv 1803 . 2  |-  (EXMID  ->  A. x A. y ( ( x  ~<_  y  /\  y  ~<_  x )  ->  x  ~~  y ) )
4 exmidsbthr 11868 . 2  |-  ( A. x A. y ( ( x  ~<_  y  /\  y  ~<_  x )  ->  x  ~~  y )  -> EXMID )
53, 4impbii 124 1  |-  (EXMID  <->  A. x A. y ( ( x  ~<_  y  /\  y  ~<_  x )  ->  x  ~~  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287   class class class wbr 3845  EXMIDwem 4029    ~~ cen 6453    ~<_ cdom 6454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-stab 776  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-exmid 4030  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-1o 6181  df-2o 6182  df-map 6405  df-en 6456  df-dom 6457  df-dju 6729  df-inl 6737  df-inr 6738  df-case 6773  df-omni 6788  df-nninf 6789
This theorem is referenced by: (None)
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