Theorem exmidsbth 13394

Description: The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 6863) 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-intuitionist proof at https://us.metamath.org/mpeuni/sbth.html 6863.

The reverse direction (exmidsbthr 13393) 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

Step	Hyp	Ref	Expression
1		isbth 6863	. . . 4 |- ( (EXMID /\ ( x ~<_ y /\ y ~<_ x ) ) -> x ~~ y )
2	1	ex 114	. . 3 |- (EXMID -> ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) )
3	2	alrimivv 1848	. 2 |- (EXMID -> A. x A. y ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) )
4		exmidsbthr 13393	. 2 |- ( A. x A. y ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) -> EXMID )
5	3, 4	impbii 125	1 |- (EXMID <-> A. x A. y ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1330   class class class wbr 3937  EXMIDwem 4126   ~~ cen 6640   ~<_ cdom 6641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-exmid 4127  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-1o 6321  df-2o 6322  df-map 6552  df-en 6643  df-dom 6644  df-dju 6931  df-inl 6940  df-inr 6941  df-case 6977  df-omni 7014  df-nninf 7015

This theorem is referenced by: (None)