Theorem axnulALT 5223

Description: Alternate proof of axnul 5224, proved from propositional calculus, ax-gen 1799, ax-4 1813, sp 2178, and ax-rep 5205. To check this, replace sp 2178 with the obsolete axiom ax-c5 36824 in the proof of axnulALT 5223 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 5205	. . 3 ⊢ (∀𝑤∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
2		sp 2178	. . . . . 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 1784	. . . . 5 ⊢ (∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥))
5	3, 4	sylibr 233	. . . 4 ⊢ (∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥) → ∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥))
6		fal 1553	. . . . . 6 ⊢ ¬ ⊥
7		sp 2178	. . . . . 6 ⊢ (∀𝑥⊥ → ⊥)
8	6, 7	mto 196	. . . . 5 ⊢ ¬ ∀𝑥⊥
9	8	pm2.21i 119	. . . 4 ⊢ (∀𝑥⊥ → 𝑦 = 𝑥)
10	5, 9	mpg 1801	. . 3 ⊢ ∃𝑥∀𝑦(∀𝑥⊥ → 𝑦 = 𝑥)
11	1, 10	mpg 1801	. 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥))
12	8	intnan 486	. . . . . 6 ⊢ ¬ (𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)
13	12	nex 1804	. . . . 5 ⊢ ¬ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)
14	13	nbn 372	. . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
15	14	albii 1823	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
16	15	exbii 1851	. 2 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑤(𝑤 ∈ 𝑧 ∧ ∀𝑥⊥)))
17	11, 16	mpbir 230	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539  ⊥wfal 1551  ∃wex 1783   ∈ wcel 2108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173  ax-rep 5205

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784

This theorem is referenced by: (None)