Theorem exmidsbth 14543

Description: The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 6961) 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 6961.

The reverse direction (exmidsbthr 14542) 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 6961	. . . 4 |- ( (EXMID /\ ( x ~<_ y /\ y ~<_ x ) ) -> x ~~ y )
2	1	ex 115	. . 3 |- (EXMID -> ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) )
3	2	alrimivv 1875	. 2 |- (EXMID -> A. x A. y ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) )
4		exmidsbthr 14542	. 2 |- ( A. x A. y ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) -> EXMID )
5	3, 4	impbii 126	1 |- (EXMID <-> A. x A. y ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1351   class class class wbr 4001  EXMIDwem 4192   ~~ cen 6733   ~<_ cdom 6734

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 4116  ax-sep 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-iinf 4585

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-iun 3887  df-br 4002  df-opab 4063  df-mpt 4064  df-tr 4100  df-exmid 4193  df-id 4291  df-iord 4364  df-on 4366  df-suc 4369  df-iom 4588  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-ov 5873  df-oprab 5874  df-mpo 5875  df-1st 6136  df-2nd 6137  df-1o 6412  df-2o 6413  df-map 6645  df-en 6736  df-dom 6737  df-dju 7032  df-inl 7041  df-inr 7042  df-case 7078  df-nninf 7114  df-omni 7128

This theorem is referenced by: (None)