Theorem pw1dceq 16827

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 4311	. . . 4 ⊢ (EXMID → DECID 𝑥 = 𝑦)
2	1	ralrimivw 2618	. . 3 ⊢ (EXMID → ∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)
3	2	ralrimivw 2618	. 2 ⊢ (EXMID → ∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦)
4		1oex 6657	. . . . . 6 ⊢ 1o ∈ V
5	4	pwid 3689	. . . . 5 ⊢ 1o ∈ 𝒫 1o
6		eqeq2 2244	. . . . . . 7 ⊢ (𝑦 = 1o → (𝑥 = 𝑦 ↔ 𝑥 = 1o))
7	6	dcbid 846	. . . . . 6 ⊢ (𝑦 = 1o → (DECID 𝑥 = 𝑦 ↔ DECID 𝑥 = 1o))
8	7	rspcv 2919	. . . . 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 2607	. . 3 ⊢ (∀𝑥 ∈ 𝒫 1o∀𝑦 ∈ 𝒫 1oDECID 𝑥 = 𝑦 → ∀𝑥 ∈ 𝒫 1oDECID 𝑥 = 1o)
11		pw1dc1 7176	. . 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 842   = wceq 1398   ∈ wcel 2205  ∀wral 2522  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-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556

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-rex 2528  df-rab 2531  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-pr 3698  df-uni 3917  df-tr 4211  df-exmid 4310  df-iord 4489  df-on 4491  df-suc 4494  df-1o 6649

This theorem is referenced by: (None)