Theorem axnulALT2 35452

Description: Alternate proof of axnul 5269, proved from propositional calculus, ax-gen 1825, ax-4 1839, ax-6 1997, and ax-rep 5239. (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 5239	. . 3 ⊢ (∀𝑤∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
2		fal 1584	. . . . . . 7 ⊢ ¬ ⊥
3	2	spfalw 2010	. . . . . . 7 ⊢ (∀𝑥⊥ → ⊥)
4	2, 3	mto 200	. . . . . 6 ⊢ ¬ ∀𝑥⊥
5	4	pm2.21i 120	. . . . 5 ⊢ (∀𝑥⊥ → 𝑦 = 𝑥)
6	5	ax-gen 1825	. . . 4 ⊢ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)
7	6	exgen 2004	. . 3 ⊢ ∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)
8	1, 7	mpg 1827	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))
9	4	intnan 491	. . . . . 6 ⊢ ¬ (𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)
10	9	nex 1830	. . . . 5 ⊢ ¬ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)
11	10	nbn 375	. . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
12	11	albii 1849	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
13	12	exbii 1878	. 2 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
14	8, 13	mpbir 234	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568  ⊥wfal 1582  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-6 1997  ax-rep 5239

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810

This theorem is referenced by: (None)