Theorem pw0ss 15848

Description: There are no inhabited subsets of the empty set. (Contributed by Jim Kingdon, 31-Dec-2025.)

Assertion

Ref	Expression
pw0ss	⊢ {𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅

Distinct variable group:   𝑗,𝑠

Proof of Theorem pw0ss

Step	Hyp	Ref	Expression
1		pw0 3794	. . 3 ⊢ 𝒫 ∅ = {∅}
2	1	rabeqi 2772	. 2 ⊢ {𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = {𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠}
3		rabeq0 3501	. . 3 ⊢ ({𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅ ↔ ∀𝑠 ∈ {∅} ¬ ∃𝑗 𝑗 ∈ 𝑠)
4		elsni 3664	. . . 4 ⊢ (𝑠 ∈ {∅} → 𝑠 = ∅)
5		notm0 3492	. . . 4 ⊢ (¬ ∃𝑗 𝑗 ∈ 𝑠 ↔ 𝑠 = ∅)
6	4, 5	sylibr 134	. . 3 ⊢ (𝑠 ∈ {∅} → ¬ ∃𝑗 𝑗 ∈ 𝑠)
7	3, 6	mprgbir 2568	. 2 ⊢ {𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅
8	2, 7	eqtri 2230	1 ⊢ {𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅

Syntax hints:  ¬ wn 3   = wceq 1375  ∃wex 1518   ∈ wcel 2180  {crab 2492  ∅c0 3471  𝒫 cpw 3629  {csn 3646

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rab 2497  df-v 2781  df-dif 3179  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652

This theorem is referenced by:  uhgr0vb  15849  uhgr0  15850