Theorem pw1dc1 7176

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 7173	. 2 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)
2		elpwi 3680	. . . . 5 ⊢ (𝑥 ∈ 𝒫 1o → 𝑥 ⊆ 1o)
3		ss1o0el1o 7175	. . . . 5 ⊢ (𝑥 ⊆ 1o → (∅ ∈ 𝑥 ↔ 𝑥 = 1o))
4	2, 3	syl 14	. . . 4 ⊢ (𝑥 ∈ 𝒫 1o → (∅ ∈ 𝑥 ↔ 𝑥 = 1o))
5	4	dcbid 846	. . 3 ⊢ (𝑥 ∈ 𝒫 1o → (DECID ∅ ∈ 𝑥 ↔ DECID 𝑥 = 1o))
6	5	ralbiia 2558	. 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 842   = wceq 1398   ∈ wcel 2205  ∀wral 2522   ⊆ wss 3213  ∅c0 3510  𝒫 cpw 3671  EXMIDwem 4309  1_oc1o 6642

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-nul 4238

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-exmid 4310  df-suc 4494  df-1o 6649

This theorem is referenced by:  pw1dceq  16827