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Mirrors > Home > ILE Home > Th. List > Mathboxes > exmidsbth | GIF version |
Description: The Schroeder-Bernstein
Theorem is equivalent to excluded middle. This
is Metamath 100 proof #25. The forward direction (isbth 6807) 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-intuitionist proof at
https://us.metamath.org/mpeuni/sbth.html 6807.
The reverse direction (exmidsbthr 12910) 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 6807 | . . . 4 ⊢ ((EXMID ∧ (𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥)) → 𝑥 ≈ 𝑦) | |
2 | 1 | ex 114 | . . 3 ⊢ (EXMID → ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦)) |
3 | 2 | alrimivv 1829 | . 2 ⊢ (EXMID → ∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦)) |
4 | exmidsbthr 12910 | . 2 ⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID) | |
5 | 3, 4 | impbii 125 | 1 ⊢ (EXMID ↔ ∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1312 class class class wbr 3895 EXMIDwem 4078 ≈ cen 6586 ≼ cdom 6587 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-coll 4003 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-iinf 4462 |
This theorem depends on definitions: df-bi 116 df-stab 799 df-dc 803 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-if 3441 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-tr 3987 df-exmid 4079 df-id 4175 df-iord 4248 df-on 4250 df-suc 4253 df-iom 4465 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-1st 5992 df-2nd 5993 df-1o 6267 df-2o 6268 df-map 6498 df-en 6589 df-dom 6590 df-dju 6875 df-inl 6884 df-inr 6885 df-case 6921 df-omni 6956 df-nninf 6957 |
This theorem is referenced by: (None) |
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