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| 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.) |
| Ref | Expression |
|---|---|
| exmidsbth | ⊢ (EXMID ↔ ∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isbth 7033 | . . . 4 ⊢ ((EXMID ∧ (𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥)) → 𝑥 ≈ 𝑦) | |
| 2 | 1 | ex 115 | . . 3 ⊢ (EXMID → ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦)) |
| 3 | 2 | alrimivv 1889 | . 2 ⊢ (EXMID → ∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦)) |
| 4 | exmidsbthr 15647 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID) | |
| 5 | 3, 4 | impbii 126 | 1 ⊢ (EXMID ↔ ∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦)) |
| 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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