Theorem ax6vsep 5257

Description: Derive ax6v 2001 (a weakened version of ax-6 2000 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5249 and Extensionality ax-ext 2733. See ax6 2414 for the derivation of ax-6 2000 from ax6v 2001. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax6vsep	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem ax6vsep

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-sep 5249	. . 3 ⊢ ∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧)))
2		id 23	. . . . . . . . 9 ⊢ (𝑧 = 𝑧 → 𝑧 = 𝑧)
3	2	biantru 539	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧)))
4	3	bibi2i 340	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧))))
5	4	biimpri 231	. . . . . 6 ⊢ ((𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧))) → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
6	5	alimi 1844	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧))) → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
7		ax-ext 2733	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
8	6, 7	syl 18	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧))) → 𝑥 = 𝑦)
9	8	eximi 1868	. . 3 ⊢ (∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧))) → ∃𝑥 𝑥 = 𝑦)
10	1, 9	ax-mp 5	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
11		df-ex 1813	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
12	10, 11	mpbi 233	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-ext 2733  ax-sep 5249

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)