Theorem zfinf 9680

Description: Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)

Assertion

Ref	Expression
zfinf	⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem zfinf

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-inf 9679	. 2 ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
2		elequ1 2114	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
3		elequ1 2114	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
4	3	anbi1d 631	. . . . . . 7 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
5	4	exbidv 1920	. . . . . 6 ⊢ (𝑤 = 𝑦 → (∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
6	2, 5	imbi12d 344	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) ↔ (𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
7	6	cbvalvw 2034	. . . 4 ⊢ (∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))
8	7	anbi2i 623	. . 3 ⊢ ((𝑦 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
9	8	exbii 1847	. 2 ⊢ (∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))))
10	1, 9	mpbi 230	1 ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537  ∃wex 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-inf 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779

This theorem is referenced by:  axinf2  9681  axinfndlem1  10646