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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 6889 | . 2 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥) | |
2 | elpwi 3575 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 1o → 𝑥 ⊆ 1o) | |
3 | ss1o0el1o 6890 | . . . . 5 ⊢ (𝑥 ⊆ 1o → (∅ ∈ 𝑥 ↔ 𝑥 = 1o)) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝑥 ∈ 𝒫 1o → (∅ ∈ 𝑥 ↔ 𝑥 = 1o)) |
5 | 4 | dcbid 833 | . . 3 ⊢ (𝑥 ∈ 𝒫 1o → (DECID ∅ ∈ 𝑥 ↔ DECID 𝑥 = 1o)) |
6 | 5 | ralbiia 2484 | . 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 829 = wceq 1348 ∈ wcel 2141 ∀wral 2448 ⊆ wss 3121 ∅c0 3414 𝒫 cpw 3566 EXMIDwem 4180 1oc1o 6388 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-nul 4115 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-exmid 4181 df-suc 4356 df-1o 6395 |
This theorem is referenced by: (None) |
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