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Mirrors > Home > ILE Home > Th. List > exmidlpo | GIF version |
Description: Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.) |
Ref | Expression |
---|---|
exmidlpo | ⊢ (EXMID → ω ∈ Omni) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidomni 6961 | . 2 ⊢ (EXMID ↔ ∀𝑥 𝑥 ∈ Omni) | |
2 | omex 4465 | . . 3 ⊢ ω ∈ V | |
3 | eleq1 2175 | . . 3 ⊢ (𝑥 = ω → (𝑥 ∈ Omni ↔ ω ∈ Omni)) | |
4 | 2, 3 | spcv 2748 | . 2 ⊢ (∀𝑥 𝑥 ∈ Omni → ω ∈ Omni) |
5 | 1, 4 | sylbi 120 | 1 ⊢ (EXMID → ω ∈ Omni) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1310 ∈ wcel 1461 EXMIDwem 4076 ωcom 4462 Omnicomni 6951 |
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 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-iinf 4460 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-opab 3948 df-mpt 3949 df-exmid 4077 df-id 4173 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fv 5087 df-1o 6264 df-2o 6265 df-omni 6953 |
This theorem is referenced by: exmidmp 6977 |
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