Theorem 0elaxnul 45166

Description: A class that contains the empty set models the Null Set Axiom ax-nul 5249. (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 4288	. . 3 ⊢ ¬ 𝑦 ∈ ∅
2	1	rgenw 3053	. 2 ⊢ ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ ∅
3		eleq2 2823	. . . . 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 3574	. 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 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  ∅c0 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-dif 3902  df-nul 4284

This theorem is referenced by:  wfaxnul  45179