Theorem pw0ss 15723

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 3782	. . 3 ⊢ 𝒫 ∅ = {∅}
2	1	rabeqi 2766	. 2 ⊢ {𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = {𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠}
3		rabeq0 3491	. . 3 ⊢ ({𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅ ↔ ∀𝑠 ∈ {∅} ¬ ∃𝑗 𝑗 ∈ 𝑠)
4		elsni 3652	. . . 4 ⊢ (𝑠 ∈ {∅} → 𝑠 = ∅)
5		notm0 3482	. . . 4 ⊢ (¬ ∃𝑗 𝑗 ∈ 𝑠 ↔ 𝑠 = ∅)
6	4, 5	sylibr 134	. . 3 ⊢ (𝑠 ∈ {∅} → ¬ ∃𝑗 𝑗 ∈ 𝑠)
7	3, 6	mprgbir 2565	. 2 ⊢ {𝑠 ∈ {∅} ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅
8	2, 7	eqtri 2227	1 ⊢ {𝑠 ∈ 𝒫 ∅ ∣ ∃𝑗 𝑗 ∈ 𝑠} = ∅

Syntax hints:  ¬ wn 3   = wceq 1373  ∃wex 1516   ∈ wcel 2177  {crab 2489  ∅c0 3461  𝒫 cpw 3617  {csn 3634

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rab 2494  df-v 2775  df-dif 3169  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640

This theorem is referenced by:  uhgr0vb  15724  uhgr0  15725