Theorem zfinf2 4312

Description: A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.)

Assertion

Ref	Expression
zfinf2	⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦

Proof of Theorem zfinf2

Step	Hyp	Ref	Expression
1		ax-iinf 4311	. 2 ⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
2		df-ral 2311	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
3	2	anbi2i 430	. . 3 ⊢ ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) ↔ (∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)))
4	3	exbii 1496	. 2 ⊢ (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥) ↔ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)))
5	1, 4	mpbir 134	1 ⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  ∅c0 3224  suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-ral 2311

This theorem is referenced by:  omex  4316