Theorem pw1dc1 7084

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 7081	. 2 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)
2		elpwi 3658	. . . . 5 ⊢ (𝑥 ∈ 𝒫 1o → 𝑥 ⊆ 1o)
3		ss1o0el1o 7083	. . . . 5 ⊢ (𝑥 ⊆ 1o → (∅ ∈ 𝑥 ↔ 𝑥 = 1o))
4	2, 3	syl 14	. . . 4 ⊢ (𝑥 ∈ 𝒫 1o → (∅ ∈ 𝑥 ↔ 𝑥 = 1o))
5	4	dcbid 843	. . 3 ⊢ (𝑥 ∈ 𝒫 1o → (DECID ∅ ∈ 𝑥 ↔ DECID 𝑥 = 1o))
6	5	ralbiia 2544	. 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 839   = wceq 1395   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197  ∅c0 3491  𝒫 cpw 3649  EXMIDwem 4278  1_oc1o 6561

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-nul 4210

This theorem depends on definitions:  df-bi 117  df-dc 840  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-exmid 4279  df-suc 4462  df-1o 6568

This theorem is referenced by:  pw1dceq  16399