Theorem pw0ss 15891

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 3815	. . 3 ⊢ 𝒫 ∅ = {∅}
2	1	rabeqi 2792	. 2 ⊢ {𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = {𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠}
3		rabeq0 3521	. . 3 ⊢ ({𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅ ↔ ∀𝑠 ∈ {∅} ¬ ∃𝑗 𝑗 ∈ 𝑠)
4		elsni 3684	. . . 4 ⊢ (𝑠 ∈ {∅} → 𝑠 = ∅)
5		notm0 3512	. . . 4 ⊢ (¬ ∃𝑗 𝑗 ∈ 𝑠 ↔ 𝑠 = ∅)
6	4, 5	sylibr 134	. . 3 ⊢ (𝑠 ∈ {∅} → ¬ ∃𝑗 𝑗 ∈ 𝑠)
7	3, 6	mprgbir 2588	. 2 ⊢ {𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅
8	2, 7	eqtri 2250	1 ⊢ {𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅

Syntax hints:  ¬ wn 3   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {crab 2512  ∅c0 3491  𝒫 cpw 3649  {csn 3666

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-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672

This theorem is referenced by:  uhgr0vb  15892  uhgr0  15893