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Mirrors > Home > ILE Home > Th. List > pw1dc1 | GIF version |
Description: If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.) |
Ref | Expression |
---|---|
pw1dc1 | ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw1dc0el 6853 | . 2 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥) | |
2 | elpwi 3552 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 1o → 𝑥 ⊆ 1o) | |
3 | ss1o0el1o 6854 | . . . . 5 ⊢ (𝑥 ⊆ 1o → (∅ ∈ 𝑥 ↔ 𝑥 = 1o)) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝑥 ∈ 𝒫 1o → (∅ ∈ 𝑥 ↔ 𝑥 = 1o)) |
5 | 4 | dcbid 824 | . . 3 ⊢ (𝑥 ∈ 𝒫 1o → (DECID ∅ ∈ 𝑥 ↔ DECID 𝑥 = 1o)) |
6 | 5 | ralbiia 2471 | . 2 ⊢ (∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥 ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o) |
7 | 1, 6 | bitri 183 | 1 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 DECID wdc 820 = wceq 1335 ∈ wcel 2128 ∀wral 2435 ⊆ wss 3102 ∅c0 3394 𝒫 cpw 3543 EXMIDwem 4155 1oc1o 6353 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-nul 4090 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-exmid 4156 df-suc 4331 df-1o 6360 |
This theorem is referenced by: (None) |
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