Theorem 0elaxnul 44966

Description: A class that contains the empty set models the Null Set Axiom ax-nul 5263. (Contributed by Eric Schmidt, 19-Oct-2025.)

Assertion

Ref	Expression
0elaxnul	⊢ (∅ ∈ 𝑀 → ∃𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ 𝑥)

Distinct variable groups:   𝑥,𝑦   𝑥,𝑀

Allowed substitution hint:   𝑀(𝑦)

Proof of Theorem 0elaxnul

Step	Hyp	Ref	Expression
1		noel 4303	. . 3 ⊢ ¬ 𝑦 ∈ ∅
2	1	rgenw 3049	. 2 ⊢ ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ ∅
3		eleq2 2818	. . . . 5 ⊢ (𝑥 = ∅ → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ∅))
4	3	notbid 318	. . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑦 ∈ 𝑥 ↔ ¬ 𝑦 ∈ ∅))
5	4	ralbidv 3157	. . 3 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ ∅))
6	5	rspcev 3591	. 2 ⊢ ((∅ ∈ 𝑀 ∧ ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ ∅) → ∃𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ 𝑥)
7	2, 6	mpan2 691	1 ⊢ (∅ ∈ 𝑀 → ∃𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  ∅c0 4298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-dif 3919  df-nul 4299

This theorem is referenced by:  wfaxnul  44979