Theorem pw1dceq 16626

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 4286	. . . 4 ⊢ (EXMID → DECID 𝑥 = 𝑦)
2	1	ralrimivw 2606	. . 3 ⊢ (EXMID → ∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)
3	2	ralrimivw 2606	. 2 ⊢ (EXMID → ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)
4		1oex 6590	. . . . . 6 ⊢ 1o ∈ V
5	4	pwid 3667	. . . . 5 ⊢ 1o ∈ 𝒫 1o
6		eqeq2 2241	. . . . . . 7 ⊢ (𝑦 = 1o → (𝑥 = 𝑦 ↔ 𝑥 = 1o))
7	6	dcbid 845	. . . . . 6 ⊢ (𝑦 = 1o → (DECID 𝑥 = 𝑦 ↔ DECID 𝑥 = 1o))
8	7	rspcv 2906	. . . . 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 2595	. . 3 ⊢ (∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦 → ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)
11		pw1dc1 7106	. . 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 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510  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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

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-rex 2516  df-rab 2519  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-pr 3676  df-uni 3894  df-tr 4188  df-exmid 4285  df-iord 4463  df-on 4465  df-suc 4468  df-1o 6582

This theorem is referenced by: (None)