Theorem exmidnotnotr 16828

Description: Excluded middle is equivalent to double negation elimination. Read an element of 𝒫 1_o as being a truth value and 𝑥 = 1_o being that 𝑥 is true. For a similar theorem, but expressed in terms of formulas rather than subsets of 1_o, see dcfromnotnotr 1493. (Contributed by Jim Kingdon, 22-Apr-2026.)

Assertion

Ref	Expression
exmidnotnotr	⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o))

Proof of Theorem exmidnotnotr

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidexmid 4311	. . . 4 ⊢ (EXMID → DECID 𝑥 = 1o)
2		notnotrdc 851	. . . 4 ⊢ (DECID 𝑥 = 1o → (¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
3	1, 2	syl 14	. . 3 ⊢ (EXMID → (¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
4	3	ralrimivw 2618	. 2 ⊢ (EXMID → ∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
5		eqeq1 2241	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 = 1o ↔ 𝑦 = 1o))
6	5	notbid 673	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 = 1o ↔ ¬ 𝑦 = 1o))
7	6	notbid 673	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (¬ ¬ 𝑥 = 1o ↔ ¬ ¬ 𝑦 = 1o))
8	7, 5	imbi12d 234	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((¬ ¬ 𝑥 = 1o → 𝑥 = 1o) ↔ (¬ ¬ 𝑦 = 1o → 𝑦 = 1o)))
9		simpl 109	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o) ∧ 𝑦 ⊆ {∅}) → ∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
10		velpw 3678	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 1o ↔ 𝑦 ⊆ 1o)
11		df1o2 6663	. . . . . . . . 9 ⊢ 1o = {∅}
12	11	sseq2i 3267	. . . . . . . 8 ⊢ (𝑦 ⊆ 1o ↔ 𝑦 ⊆ {∅})
13	10, 12	sylbbr 136	. . . . . . 7 ⊢ (𝑦 ⊆ {∅} → 𝑦 ∈ 𝒫 1o)
14	13	adantl 277	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o) ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ 𝒫 1o)
15	8, 9, 14	rspcdva 2928	. . . . 5 ⊢ ((∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o) ∧ 𝑦 ⊆ {∅}) → (¬ ¬ 𝑦 = 1o → 𝑦 = 1o))
16		df-stab 839	. . . . 5 ⊢ (STAB 𝑦 = 1o ↔ (¬ ¬ 𝑦 = 1o → 𝑦 = 1o))
17	15, 16	sylibr 134	. . . 4 ⊢ ((∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o) ∧ 𝑦 ⊆ {∅}) → STAB 𝑦 = 1o)
18	11	eqeq2i 2245	. . . . . 6 ⊢ (𝑦 = 1o ↔ 𝑦 = {∅})
19	18	a1i 9	. . . . 5 ⊢ ((∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o) ∧ 𝑦 ⊆ {∅}) → (𝑦 = 1o ↔ 𝑦 = {∅}))
20	19	stbid 840	. . . 4 ⊢ ((∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o) ∧ 𝑦 ⊆ {∅}) → (STAB 𝑦 = 1o ↔ STAB 𝑦 = {∅}))
21	17, 20	mpbid 147	. . 3 ⊢ ((∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o) ∧ 𝑦 ⊆ {∅}) → STAB 𝑦 = {∅})
22	21	exmid1stab 4323	. 2 ⊢ (∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o) → EXMID)
23	4, 22	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  STAB wstab 838  DECID wdc 842   = wceq 1398   ∈ wcel 2205  ∀wral 2522   ⊆ wss 3213  ∅c0 3510  𝒫 cpw 3671  {csn 3691  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-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  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-exmid 4310  df-suc 4494  df-1o 6649

This theorem is referenced by:  exmidcon  16829  exmidpeirce  16830