Theorem exmidnotnotr 16666

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 1492. (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 4288	. . . 4 ⊢ (EXMID → DECID 𝑥 = 1o)
2		notnotrdc 850	. . . 4 ⊢ (DECID 𝑥 = 1o → (¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
3	1, 2	syl 14	. . 3 ⊢ (EXMID → (¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
4	3	ralrimivw 2605	. 2 ⊢ (EXMID → ∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o))
5		eqeq1 2237	. . . . . . . . 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 3660	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 1o ↔ 𝑦 ⊆ 1o)
11		df1o2 6601	. . . . . . . . 9 ⊢ 1o = {∅}
12	11	sseq2i 3253	. . . . . . . 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 2914	. . . . 5 ⊢ ((∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o) ∧ 𝑦 ⊆ {∅}) → (¬ ¬ 𝑦 = 1o → 𝑦 = 1o))
16		df-stab 838	. . . . 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 2241	. . . . . 6 ⊢ (𝑦 = 1o ↔ 𝑦 = {∅})
19	18	a1i 9	. . . . 5 ⊢ ((∀𝑥 ∈ 𝒫 1o(¬ ¬ 𝑥 = 1o → 𝑥 = 1o) ∧ 𝑦 ⊆ {∅}) → (𝑦 = 1o ↔ 𝑦 = {∅}))
20	19	stbid 839	. . . 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 4300	. 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 837  DECID wdc 841   = wceq 1397   ∈ wcel 2201  ∀wral 2509   ⊆ wss 3199  ∅c0 3493  𝒫 cpw 3653  {csn 3670  EXMIDwem 4286  1_oc1o 6580

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-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rab 2518  df-v 2803  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-exmid 4287  df-suc 4470  df-1o 6587

This theorem is referenced by:  exmidcon  16667  exmidpeirce  16668