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Theorem zfinf2 4510
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 4509 . 2 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥))
2 df-ral 2422 . . . 4 (∀𝑦𝑥 suc 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥))
32anbi2i 453 . . 3 ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ (∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)))
43exbii 1585 . 2 (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥) ↔ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)))
51, 4mpbir 145 1 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1330  wex 1469  wcel 1481  wral 2417  c0 3367  suc csuc 4294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-ral 2422
This theorem is referenced by:  omex  4514
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