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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 7014 | . 2 ⊢ (EXMID ↔ ∀𝑥 𝑥 ∈ Omni) | |
2 | omex 4507 | . . 3 ⊢ ω ∈ V | |
3 | eleq1 2202 | . . 3 ⊢ (𝑥 = ω → (𝑥 ∈ Omni ↔ ω ∈ Omni)) | |
4 | 2, 3 | spcv 2779 | . 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 1329 ∈ wcel 1480 EXMIDwem 4118 ωcom 4504 Omnicomni 7004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-exmid 4119 df-id 4215 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-1o 6313 df-2o 6314 df-omni 7006 |
This theorem is referenced by: exmidmp 7031 |
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