Theorem 0elaxnul 45807

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 4284	. . 3 ⊢ ¬ 𝑦 ∈ ∅
2	1	rgenw 3080	. 2 ⊢ ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ ∅
3		eleq2 2849	. . . . 5 ⊢ (𝑥 = ∅ → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ∅))
4	3	notbid 321	. . . 4 ⊢ (𝑥 = ∅ → (¬ 𝑦 ∈ 𝑥 ↔ ¬ 𝑦 ∈ ∅))
5	4	ralbidv 3185	. . 3 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ ∅))
6	5	rspcev 3576	. 2 ⊢ ((∅ ∈ 𝑀 ∧ ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ ∅) → ∃𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ 𝑥)
7	2, 6	mpan2 704	1 ⊢ (∅ ∈ 𝑀 → ∃𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ 𝑥)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  ∅c0 4279

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-dif 3902  df-nul 4280

This theorem is used by:  wfaxnul  45820