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Theorem exmidsbth 15648
Description: The Schroeder-Bernstein Theorem is equivalent to excluded middle. This is Metamath 100 proof #25. The forward direction (isbth 7033) 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 7033.

The reverse direction (exmidsbthr 15647) 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 7033 . . . 4 ((EXMID ∧ (𝑥𝑦𝑦𝑥)) → 𝑥𝑦)
21ex 115 . . 3 (EXMID → ((𝑥𝑦𝑦𝑥) → 𝑥𝑦))
32alrimivv 1889 . 2 (EXMID → ∀𝑥𝑦((𝑥𝑦𝑦𝑥) → 𝑥𝑦))
4 exmidsbthr 15647 . 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 4033  EXMIDwem 4227  cen 6797  cdom 6798
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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624
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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-exmid 4228  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-1o 6474  df-2o 6475  df-map 6709  df-en 6800  df-dom 6801  df-dju 7104  df-inl 7113  df-inr 7114  df-case 7150  df-nninf 7186  df-omni 7201
This theorem is referenced by: (None)
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