Theorem pw1dc1 6970

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.)

Assertion

Ref	Expression
pw1dc1	⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)

Proof of Theorem pw1dc1

Step	Hyp	Ref	Expression
1		pw1dc0el 6967	. 2 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)
2		elpwi 3610	. . . . 5 ⊢ (𝑥 ∈ 𝒫 1o → 𝑥 ⊆ 1o)
3		ss1o0el1o 6969	. . . . 5 ⊢ (𝑥 ⊆ 1o → (∅ ∈ 𝑥 ↔ 𝑥 = 1o))
4	2, 3	syl 14	. . . 4 ⊢ (𝑥 ∈ 𝒫 1o → (∅ ∈ 𝑥 ↔ 𝑥 = 1o))
5	4	dcbid 839	. . 3 ⊢ (𝑥 ∈ 𝒫 1o → (DECID ∅ ∈ 𝑥 ↔ DECID 𝑥 = 1o))
6	5	ralbiia 2508	. 2 ⊢ (∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥 ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)
7	1, 6	bitri 184	1 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)

Syntax hints:   ↔ wb 105  DECID wdc 835   = wceq 1364   ∈ wcel 2164  ∀wral 2472   ⊆ wss 3153  ∅c0 3446  𝒫 cpw 3601  EXMIDwem 4223  1_oc1o 6462

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-ext 2175  ax-nul 4155

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-exmid 4224  df-suc 4402  df-1o 6469

This theorem is referenced by: (None)