Theorem zfinf2 4255

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 4254	. 2 ⊢ ∃x(∅ ∈ x ∧ ∀y(y ∈ x → suc y ∈ x))
2		df-ral 2305	. . . 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 1493	. 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 1240  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  ∅c0 3218  suc csuc 4068

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-ral 2305

This theorem is referenced by:  omex  4259