Theorem pw1m 7534

Description: A truth value which is inhabited is equal to true. This is a variation of pwntru 4312 and pwtrufal 16771. (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 3678	. . . . . . . 8 ⊢ (𝐴 ∈ 𝒫 1o → 𝐴 ⊆ 1o)
2		df1o2 6661	. . . . . . . 8 ⊢ 1o = {∅}
3	1, 2	sseqtrdi 3286	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 1o → 𝐴 ⊆ {∅})
4	3	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ {∅})
5	1	sselda 3238	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o)
6	5, 2	eleqtrdi 2325	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {∅})
7		elsni 3707	. . . . . . . . 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 2310	. . . . . . 7 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → ∅ ∈ 𝐴)
11	10	snssd 3839	. . . . . 6 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → {∅} ⊆ 𝐴)
12	4, 11	eqssd 3255	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝐴 = {∅})
13	12, 2	eqtr4di 2283	. . . 4 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝐴 = 1o)
14	13	ex 115	. . 3 ⊢ (𝐴 ∈ 𝒫 1o → (𝑥 ∈ 𝐴 → 𝐴 = 1o))
15	14	exlimdv 1868	. 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 1398  ∃wex 1541   ∈ wcel 2203   ⊆ wss 3211  ∅c0 3508  𝒫 cpw 3669  {csn 3689  1_oc1o 6640

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-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-suc 4492  df-1o 6647

This theorem is referenced by:  pw1if  7535