Theorem pw1m 7355

Description: A truth value which is inhabited is equal to true. This is a variation of pwntru 4251 and pwtrufal 16075. (Contributed by Jim Kingdon, 10-Jan-2026.)

Assertion

Ref	Expression
pw1m	⊢ ((𝐴 ∈ 𝒫 1o ∧ ∃𝑥 𝑥 ∈ 𝐴) → 𝐴 = 1o)

Distinct variable group:   𝑥,𝐴

Proof of Theorem pw1m

Step	Hyp	Ref	Expression
1		elpwi 3630	. . . . . . . 8 ⊢ (𝐴 ∈ 𝒫 1o → 𝐴 ⊆ 1o)
2		df1o2 6528	. . . . . . . 8 ⊢ 1o = {∅}
3	1, 2	sseqtrdi 3245	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 1o → 𝐴 ⊆ {∅})
4	3	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ {∅})
5	1	sselda 3197	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o)
6	5, 2	eleqtrdi 2299	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {∅})
7		elsni 3656	. . . . . . . . 9 ⊢ (𝑥 ∈ {∅} → 𝑥 = ∅)
8	6, 7	syl 14	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅)
9		simpr 110	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
10	8, 9	eqeltrrd 2284	. . . . . . 7 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → ∅ ∈ 𝐴)
11	10	snssd 3784	. . . . . 6 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → {∅} ⊆ 𝐴)
12	4, 11	eqssd 3214	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝐴 = {∅})
13	12, 2	eqtr4di 2257	. . . 4 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝐴 = 1o)
14	13	ex 115	. . 3 ⊢ (𝐴 ∈ 𝒫 1o → (𝑥 ∈ 𝐴 → 𝐴 = 1o))
15	14	exlimdv 1843	. 2 ⊢ (𝐴 ∈ 𝒫 1o → (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 = 1o))
16	15	imp 124	1 ⊢ ((𝐴 ∈ 𝒫 1o ∧ ∃𝑥 𝑥 ∈ 𝐴) → 𝐴 = 1o)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2177   ⊆ wss 3170  ∅c0 3464  𝒫 cpw 3621  {csn 3638  1_oc1o 6508

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-suc 4426  df-1o 6515

This theorem is referenced by:  pw1if  7356