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Theorem exmidsbth 15176
Description: The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 6985) 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 6985.

The reverse direction (exmidsbthr 15175) 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 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦))
Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidsbth
StepHypRef Expression
1 isbth 6985 . . . 4 ((EXMID ∧ (𝑥𝑦𝑦𝑥)) → 𝑥𝑦)
21ex 115 . . 3 (EXMID → ((𝑥𝑦𝑦𝑥) → 𝑥𝑦))
32alrimivv 1886 . 2 (EXMID → ∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦))
4 exmidsbthr 15175 . 2 (∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦) → EXMID)
53, 4impbii 126 1 (EXMID ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362   class class class wbr 4018  EXMIDwem 4209  cen 6756  cdom 6757
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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-exmid 4210  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-1o 6435  df-2o 6436  df-map 6668  df-en 6759  df-dom 6760  df-dju 7056  df-inl 7065  df-inr 7066  df-case 7102  df-nninf 7138  df-omni 7152
This theorem is referenced by: (None)
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