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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 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
Distinct variable group:   𝑥,𝑦

Proof of Theorem zfinf2
StepHypRef Expression
1 ax-iinf 4311 . 2 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥))
2 df-ral 2311 . . . 4 (∀𝑦𝑥 suc 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥))
32anbi2i 430 . . 3 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ (∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)))
43exbii 1496 . 2 (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)))
51, 4mpbir 134 1 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
 Colors of variables: wff set class 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
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