Theorem axnulALT 5251

Description: Alternate proof of axnul 5252, proved from propositional calculus, ax-gen 1797, ax-4 1811, sp 2191, and ax-rep 5226. To check this, replace sp 2191 with the obsolete axiom ax-c5 39253 in the proof of axnulALT 5251 and type the Metamath program "MM> SHOW TRACE_BACK axnulALT / AXIOMS" command. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem axnulALT

Dummy variables 𝑧 𝑤 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		sp 2191	. . . . . 6 ⊢ (∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥))
3	2	con2i 139	. . . . 5 ⊢ (∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥))
4		df-ex 1782	. . . . 5 ⊢ (∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥))
5	3, 4	sylibr 234	. . . 4 ⊢ (∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥))
6		fal 1556	. . . . . 6 ⊢ ¬ ⊥
7		sp 2191	. . . . . 6 ⊢ (∀𝑥⊥ → ⊥)
8	6, 7	mto 197	. . . . 5 ⊢ ¬ ∀𝑥⊥
9	8	pm2.21i 119	. . . 4 ⊢ (∀𝑥⊥ → 𝑦 = 𝑥)
10	5, 9	mpg 1799	. . 3 ⊢ ∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)
11	1, 10	mpg 1799	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))
12	8	intnan 486	. . . . . 6 ⊢ ¬ (𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)
13	12	nex 1802	. . . . 5 ⊢ ¬ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)
14	13	nbn 372	. . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
15	14	albii 1821	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
16	15	exbii 1850	. 2 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
17	11, 16	mpbir 231	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  ⊥wfal 1554  ∃wex 1781   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185  ax-rep 5226

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782

This theorem is referenced by: (None)