Theorem stnot 17039

Description: A proposition is double negation stable if and only if it is equivalent to a negated proposition. Here by "proposition" we mean a subset of a singleton (which is a choice which allows us to quantify over them). Posed as an exercise online by Yannick Forster. (Contributed by Jim Kingdon, 24-Jul-2026.)

Assertion

Ref	Expression
stnot	⊢ (𝐴 ∈ 𝒫 1o → ((¬ ¬ 𝐴 = 1o → 𝐴 = 1o) ↔ ∃𝑦 ∈ 𝒫 1o(𝐴 = 1o ↔ ¬ 𝑦 = 1o)))

Distinct variable group:   𝑦,𝐴

Proof of Theorem stnot

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2245	. . . . . 6 ⊢ (𝑦 = {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} → (𝑦 = 1o ↔ {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} = 1o))
2	1	notbid 677	. . . . 5 ⊢ (𝑦 = {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} → (¬ 𝑦 = 1o ↔ ¬ {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} = 1o))
3	2	bibi2d 232	. . . 4 ⊢ (𝑦 = {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} → ((𝐴 = 1o ↔ ¬ 𝑦 = 1o) ↔ (𝐴 = 1o ↔ ¬ {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} = 1o)))
4		1oex 6695	. . . . . 6 ⊢ 1o ∈ V
5		ssrab2 3333	. . . . . 6 ⊢ {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} ⊆ 1o
6	4, 5	elpwi2 4294	. . . . 5 ⊢ {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} ∈ 𝒫 1o
7	6	a1i 9	. . . 4 ⊢ ((𝐴 ∈ 𝒫 1o ∧ (¬ ¬ 𝐴 = 1o → 𝐴 = 1o)) → {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} ∈ 𝒫 1o)
8		notnot 638	. . . . . 6 ⊢ (𝐴 = 1o → ¬ ¬ 𝐴 = 1o)
9		simpr 110	. . . . . 6 ⊢ ((𝐴 ∈ 𝒫 1o ∧ (¬ ¬ 𝐴 = 1o → 𝐴 = 1o)) → (¬ ¬ 𝐴 = 1o → 𝐴 = 1o))
10	8, 9	impbid2 143	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 1o ∧ (¬ ¬ 𝐴 = 1o → 𝐴 = 1o)) → (𝐴 = 1o ↔ ¬ ¬ 𝐴 = 1o))
11		rabid2 2729	. . . . . . . . 9 ⊢ (1o = {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} ↔ ∀𝑥 ∈ 1o ¬ 𝐴 = 1o)
12		eqcom 2240	. . . . . . . . 9 ⊢ ({𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} = 1o ↔ 1o = {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o})
13		0lt1o 6713	. . . . . . . . . 10 ⊢ ∅ ∈ 1o
14		elex2 2838	. . . . . . . . . 10 ⊢ (∅ ∈ 1o → ∃𝑤 𝑤 ∈ 1o)
15		r19.3rmv 3618	. . . . . . . . . 10 ⊢ (∃𝑤 𝑤 ∈ 1o → (¬ 𝐴 = 1o ↔ ∀𝑥 ∈ 1o ¬ 𝐴 = 1o))
16	13, 14, 15	mp2b 8	. . . . . . . . 9 ⊢ (¬ 𝐴 = 1o ↔ ∀𝑥 ∈ 1o ¬ 𝐴 = 1o)
17	11, 12, 16	3bitr4ri 213	. . . . . . . 8 ⊢ (¬ 𝐴 = 1o ↔ {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} = 1o)
18	17	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 1o → (¬ 𝐴 = 1o ↔ {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} = 1o))
19	18	notbid 677	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 1o → (¬ ¬ 𝐴 = 1o ↔ ¬ {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} = 1o))
20	19	adantr 276	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 1o ∧ (¬ ¬ 𝐴 = 1o → 𝐴 = 1o)) → (¬ ¬ 𝐴 = 1o ↔ ¬ {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} = 1o))
21	10, 20	bitrd 188	. . . 4 ⊢ ((𝐴 ∈ 𝒫 1o ∧ (¬ ¬ 𝐴 = 1o → 𝐴 = 1o)) → (𝐴 = 1o ↔ ¬ {𝑥 ∈ 1o ∣ ¬ 𝐴 = 1o} = 1o))
22	3, 7, 21	rspcedvdw 2936	. . 3 ⊢ ((𝐴 ∈ 𝒫 1o ∧ (¬ ¬ 𝐴 = 1o → 𝐴 = 1o)) → ∃𝑦 ∈ 𝒫 1o(𝐴 = 1o ↔ ¬ 𝑦 = 1o))
23	22	ex 115	. 2 ⊢ (𝐴 ∈ 𝒫 1o → ((¬ ¬ 𝐴 = 1o → 𝐴 = 1o) → ∃𝑦 ∈ 𝒫 1o(𝐴 = 1o ↔ ¬ 𝑦 = 1o)))
24		simpr 110	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝒫 1o) ∧ (𝐴 = 1o ↔ ¬ 𝑦 = 1o)) ∧ ¬ ¬ 𝐴 = 1o) → ¬ ¬ 𝐴 = 1o)
25		simplr 533	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝒫 1o) ∧ (𝐴 = 1o ↔ ¬ 𝑦 = 1o)) ∧ ¬ ¬ 𝐴 = 1o) → (𝐴 = 1o ↔ ¬ 𝑦 = 1o))
26	25	notbid 677	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝒫 1o) ∧ (𝐴 = 1o ↔ ¬ 𝑦 = 1o)) ∧ ¬ ¬ 𝐴 = 1o) → (¬ 𝐴 = 1o ↔ ¬ ¬ 𝑦 = 1o))
27	24, 26	mtbid 683	. . . . . 6 ⊢ ((((𝐴 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝒫 1o) ∧ (𝐴 = 1o ↔ ¬ 𝑦 = 1o)) ∧ ¬ ¬ 𝐴 = 1o) → ¬ ¬ ¬ 𝑦 = 1o)
28		notnotnot 643	. . . . . 6 ⊢ (¬ ¬ ¬ 𝑦 = 1o ↔ ¬ 𝑦 = 1o)
29	27, 28	sylib 122	. . . . 5 ⊢ ((((𝐴 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝒫 1o) ∧ (𝐴 = 1o ↔ ¬ 𝑦 = 1o)) ∧ ¬ ¬ 𝐴 = 1o) → ¬ 𝑦 = 1o)
30	29, 25	mpbird 167	. . . 4 ⊢ ((((𝐴 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝒫 1o) ∧ (𝐴 = 1o ↔ ¬ 𝑦 = 1o)) ∧ ¬ ¬ 𝐴 = 1o) → 𝐴 = 1o)
31	30	ex 115	. . 3 ⊢ (((𝐴 ∈ 𝒫 1o ∧ 𝑦 ∈ 𝒫 1o) ∧ (𝐴 = 1o ↔ ¬ 𝑦 = 1o)) → (¬ ¬ 𝐴 = 1o → 𝐴 = 1o))
32	31	rexlimdva2 2671	. 2 ⊢ (𝐴 ∈ 𝒫 1o → (∃𝑦 ∈ 𝒫 1o(𝐴 = 1o ↔ ¬ 𝑦 = 1o) → (¬ ¬ 𝐴 = 1o → 𝐴 = 1o)))
33	23, 32	impbid 129	1 ⊢ (𝐴 ∈ 𝒫 1o → ((¬ ¬ 𝐴 = 1o → 𝐴 = 1o) ↔ ∃𝑦 ∈ 𝒫 1o(𝐴 = 1o ↔ ¬ 𝑦 = 1o)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532  Vcvv 2821  ∅c0 3520  𝒫 cpw 3688  1_oc1o 6680

This proof depends on 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 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  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 3715  df-pr 3716  df-uni 3936  df-tr 4230  df-iord 4511  df-on 4513  df-suc 4516  df-1o 6687

This theorem is used by: (None)