Theorem pw0ss 16307

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 3860	. . 3 ⊢ 𝒫 ∅ = {∅}
2	1	rabeqi 2814	. 2 ⊢ {𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = {𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠}
3		rabeq0 3552	. . 3 ⊢ ({𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅ ↔ ∀𝑠 ∈ {∅} ¬ ∃𝑗 𝑗 ∈ 𝑠)
4		elsni 3726	. . . 4 ⊢ (𝑠 ∈ {∅} → 𝑠 = ∅)
5		notm0 3542	. . . 4 ⊢ (¬ ∃𝑗 𝑗 ∈ 𝑠 ↔ 𝑠 = ∅)
6	4, 5	sylibr 134	. . 3 ⊢ (𝑠 ∈ {∅} → ¬ ∃𝑗 𝑗 ∈ 𝑠)
7	3, 6	mprgbir 2608	. 2 ⊢ {𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅
8	2, 7	eqtri 2259	1 ⊢ {𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅

Syntax hints:  ¬ wn 3   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {crab 2532  ∅c0 3520  𝒫 cpw 3688  {csn 3708

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 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-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714

This theorem is referenced by:  uhgr0vb  16308  uhgr0  16309