Theorem pw1dceq 16399

Description: The powerset of 1_o having decidable equality is equivalent to excluded middle. (Contributed by Jim Kingdon, 12-Feb-2026.)

Assertion

Ref	Expression
pw1dceq	⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)

Distinct variable group:   𝑥,𝑦

Proof of Theorem pw1dceq

Step	Hyp	Ref	Expression
1		exmidexmid 4280	. . . 4 ⊢ (EXMID → DECID 𝑥 = 𝑦)
2	1	ralrimivw 2604	. . 3 ⊢ (EXMID → ∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)
3	2	ralrimivw 2604	. 2 ⊢ (EXMID → ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)
4		1oex 6576	. . . . . 6 ⊢ 1o ∈ V
5	4	pwid 3664	. . . . 5 ⊢ 1o ∈ 𝒫 1o
6		eqeq2 2239	. . . . . . 7 ⊢ (𝑦 = 1o → (𝑥 = 𝑦 ↔ 𝑥 = 1o))
7	6	dcbid 843	. . . . . 6 ⊢ (𝑦 = 1o → (DECID 𝑥 = 𝑦 ↔ DECID 𝑥 = 1o))
8	7	rspcv 2903	. . . . 5 ⊢ (1o ∈ 𝒫 1o → (∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦 → DECID 𝑥 = 1o))
9	5, 8	ax-mp 5	. . . 4 ⊢ (∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦 → DECID 𝑥 = 1o)
10	9	ralimi 2593	. . 3 ⊢ (∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦 → ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)
11		pw1dc1 7084	. . 3 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)
12	10, 11	sylibr 134	. 2 ⊢ (∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦 → EXMID)
13	3, 12	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 105  DECID wdc 839   = wceq 1395   ∈ wcel 2200  ∀wral 2508  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-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

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-rex 2514  df-rab 2517  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-pr 3673  df-uni 3889  df-tr 4183  df-exmid 4279  df-iord 4457  df-on 4459  df-suc 4462  df-1o 6568

This theorem is referenced by: (None)