Theorem axnulALT2 35341

Description: Alternate proof of axnul 5254, proved from propositional calculus, ax-gen 1814, ax-4 1828, ax-6 1986, and ax-rep 5226. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BTernaryTau, 27-Mar-2026.)

Assertion

Ref	Expression
axnulALT2	⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Distinct variable group:   𝑥,𝑦

Proof of Theorem axnulALT2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-rep 5226	. . 3 ⊢ (∀𝑤∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
2		fal 1573	. . . . . . 7 ⊢ ¬ ⊥
3	2	spfalw 1999	. . . . . . 7 ⊢ (∀𝑥⊥ → ⊥)
4	2, 3	mto 199	. . . . . 6 ⊢ ¬ ∀𝑥⊥
5	4	pm2.21i 119	. . . . 5 ⊢ (∀𝑥⊥ → 𝑦 = 𝑥)
6	5	ax-gen 1814	. . . 4 ⊢ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)
7	6	exgen 1993	. . 3 ⊢ ∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)
8	1, 7	mpg 1816	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))
9	4	intnan 490	. . . . . 6 ⊢ ¬ (𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)
10	9	nex 1819	. . . . 5 ⊢ ¬ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)
11	10	nbn 374	. . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
12	11	albii 1838	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
13	12	exbii 1867	. 2 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
14	8, 13	mpbir 233	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557  ⊥wfal 1571  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-6 1986  ax-rep 5226

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799

This theorem is referenced by: (None)