Theorem pw0ss 16127

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 3843	. . 3 ⊢ 𝒫 ∅ = {∅}
2	1	rabeqi 2808	. 2 ⊢ {𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = {𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠}
3		rabeq0 3540	. . 3 ⊢ ({𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅ ↔ ∀𝑠 ∈ {∅} ¬ ∃𝑗 𝑗 ∈ 𝑠)
4		elsni 3709	. . . 4 ⊢ (𝑠 ∈ {∅} → 𝑠 = ∅)
5		notm0 3531	. . . 4 ⊢ (¬ ∃𝑗 𝑗 ∈ 𝑠 ↔ 𝑠 = ∅)
6	4, 5	sylibr 134	. . 3 ⊢ (𝑠 ∈ {∅} → ¬ ∃𝑗 𝑗 ∈ 𝑠)
7	3, 6	mprgbir 2602	. 2 ⊢ {𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅
8	2, 7	eqtri 2255	1 ⊢ {𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅

Syntax hints:  ¬ wn 3   = wceq 1398  ∃wex 1541   ∈ wcel 2205  {crab 2526  ∅c0 3510  𝒫 cpw 3671  {csn 3691

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rab 2531  df-v 2817  df-dif 3215  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697

This theorem is referenced by:  uhgr0vb  16128  uhgr0  16129