Theorem exmidsbth 15156

Description: The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 6983) 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 6983.

The reverse direction (exmidsbthr 15155) 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 6983	. . . 4 |- ( (EXMID /\ ( x ~<_ y /\ y ~<_ x ) ) -> x ~~ y )
2	1	ex 115	. . 3 |- (EXMID -> ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) )
3	2	alrimivv 1885	. 2 |- (EXMID -> A. x A. y ( ( x ~<_ y /\ y ~<_ x ) -> x ~~ y ) )
4		exmidsbthr 15155	. 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 1361   class class class wbr 4017  EXMIDwem 4208   ~~ cen 6755   ~<_ cdom 6756

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-reu 2474  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-exmid 4209  df-id 4307  df-iord 4380  df-on 4382  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-1o 6434  df-2o 6435  df-map 6667  df-en 6758  df-dom 6759  df-dju 7054  df-inl 7063  df-inr 7064  df-case 7100  df-nninf 7136  df-omni 7150

This theorem is referenced by: (None)