Theorem pw1dc1 7106

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 7103	. 2 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)
2		elpwi 3661	. . . . 5 ⊢ (𝑥 ∈ 𝒫 1o → 𝑥 ⊆ 1o)
3		ss1o0el1o 7105	. . . . 5 ⊢ (𝑥 ⊆ 1o → (∅ ∈ 𝑥 ↔ 𝑥 = 1o))
4	2, 3	syl 14	. . . 4 ⊢ (𝑥 ∈ 𝒫 1o → (∅ ∈ 𝑥 ↔ 𝑥 = 1o))
5	4	dcbid 845	. . 3 ⊢ (𝑥 ∈ 𝒫 1o → (DECID ∅ ∈ 𝑥 ↔ DECID 𝑥 = 1o))
6	5	ralbiia 2546	. 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 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510   ⊆ wss 3200  ∅c0 3494  𝒫 cpw 3652  EXMIDwem 4284  1_oc1o 6575

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-nul 4215

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-exmid 4285  df-suc 4468  df-1o 6582

This theorem is referenced by:  pw1dceq  16626