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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 7135 | . 2 ⊢ (EXMID ↔ ∀𝑥 𝑥 ∈ Omni) | |
2 | omex 4590 | . . 3 ⊢ ω ∈ V | |
3 | eleq1 2240 | . . 3 ⊢ (𝑥 = ω → (𝑥 ∈ Omni ↔ ω ∈ Omni)) | |
4 | 2, 3 | spcv 2831 | . 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 1351 ∈ wcel 2148 EXMIDwem 4192 ωcom 4587 Omnicomni 7127 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4119 ax-nul 4127 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-iinf 4585 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-int 3844 df-br 4002 df-opab 4063 df-mpt 4064 df-exmid 4193 df-id 4291 df-suc 4369 df-iom 4588 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-rn 4635 df-iota 5175 df-fun 5215 df-fn 5216 df-f 5217 df-fv 5221 df-1o 6412 df-2o 6413 df-omni 7128 |
This theorem is referenced by: exmidmp 7150 |
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