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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 7158 | . 2 ⊢ (EXMID ↔ ∀𝑥 𝑥 ∈ Omni) | |
2 | omex 4607 | . . 3 ⊢ ω ∈ V | |
3 | eleq1 2252 | . . 3 ⊢ (𝑥 = ω → (𝑥 ∈ Omni ↔ ω ∈ Omni)) | |
4 | 2, 3 | spcv 2846 | . 2 ⊢ (∀𝑥 𝑥 ∈ Omni → ω ∈ Omni) |
5 | 1, 4 | sylbi 121 | 1 ⊢ (EXMID → ω ∈ Omni) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1362 ∈ wcel 2160 EXMIDwem 4209 ωcom 4604 Omnicomni 7150 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-exmid 4210 df-id 4308 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-1o 6435 df-2o 6436 df-omni 7151 |
This theorem is referenced by: exmidmp 7173 |
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