Theorem pw1dceq 17017

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 4331	. . . 4 ⊢ (EXMID → DECID 𝑥 = 𝑦)
2	1	ralrimivw 2624	. . 3 ⊢ (EXMID → ∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)
3	2	ralrimivw 2624	. 2 ⊢ (EXMID → ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)
4		1oex 6689	. . . . . 6 ⊢ 1o ∈ V
5	4	pwid 3706	. . . . 5 ⊢ 1o ∈ 𝒫 1o
6		eqeq2 2248	. . . . . . 7 ⊢ (𝑦 = 1o → (𝑥 = 𝑦 ↔ 𝑥 = 1o))
7	6	dcbid 850	. . . . . 6 ⊢ (𝑦 = 1o → (DECID 𝑥 = 𝑦 ↔ DECID 𝑥 = 1o))
8	7	rspcv 2925	. . . . 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 2613	. . 3 ⊢ (∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦 → ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)
11		pw1dc1 7215	. . 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 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528  𝒫 cpw 3688  EXMIDwem 4329  1_oc1o 6674

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-uni 3934  df-tr 4228  df-exmid 4330  df-iord 4509  df-on 4511  df-suc 4514  df-1o 6681

This theorem is referenced by: (None)