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Axiom ax-infvn 10879
Description: Axiom of infinity in a constructive setting. This asserts the existence of the special set we want (the set of natural numbers), instead of the existence of a set with some properties (ax-iinf 4331) from which one then proves, using full separation, that the wanted set exists (omex 4336). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
Assertion
Ref Expression
ax-infvn 𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Axiom ax-infvn
StepHypRef Expression
1 vx . . . . 5 setvar 𝑥
21cv 1284 . . . 4 class 𝑥
32wind 10864 . . 3 wff Ind 𝑥
4 vy . . . . . . 7 setvar 𝑦
54cv 1284 . . . . . 6 class 𝑦
65wind 10864 . . . . 5 wff Ind 𝑦
72, 5wss 2974 . . . . 5 wff 𝑥𝑦
86, 7wi 4 . . . 4 wff (Ind 𝑦𝑥𝑦)
98, 4wal 1283 . . 3 wff 𝑦(Ind 𝑦𝑥𝑦)
103, 9wa 102 . 2 wff (Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))
1110, 1wex 1422 1 wff 𝑥(Ind 𝑥 ∧ ∀𝑦(Ind 𝑦𝑥𝑦))
Colors of variables: wff set class
This axiom is referenced by:  bj-omex  10880
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