Theorem pw1m 7405

Description: A truth value which is inhabited is equal to true. This is a variation of pwntru 4282 and pwtrufal 16322. (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 3658	. . . . . . . 8 ⊢ (𝐴 ∈ 𝒫 1o → 𝐴 ⊆ 1o)
2		df1o2 6573	. . . . . . . 8 ⊢ 1o = {∅}
3	1, 2	sseqtrdi 3272	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 1o → 𝐴 ⊆ {∅})
4	3	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ {∅})
5	1	sselda 3224	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o)
6	5, 2	eleqtrdi 2322	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ {∅})
7		elsni 3684	. . . . . . . . 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 2307	. . . . . . 7 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → ∅ ∈ 𝐴)
11	10	snssd 3812	. . . . . 6 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → {∅} ⊆ 𝐴)
12	4, 11	eqssd 3241	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝐴 = {∅})
13	12, 2	eqtr4di 2280	. . . 4 ⊢ ((𝐴 ∈ 𝒫 1o ∧ 𝑥 ∈ 𝐴) → 𝐴 = 1o)
14	13	ex 115	. . 3 ⊢ (𝐴 ∈ 𝒫 1o → (𝑥 ∈ 𝐴 → 𝐴 = 1o))
15	14	exlimdv 1865	. 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 1395  ∃wex 1538   ∈ wcel 2200   ⊆ wss 3197  ∅c0 3491  𝒫 cpw 3649  {csn 3666  1_oc1o 6553

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-suc 4461  df-1o 6560

This theorem is referenced by:  pw1if  7406