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Axiom ax-infvn 10921
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 4337) from which one then proves, using full separation, that the wanted set exists (omex 4342). "vn" is for "von Neumann". (Contributed by BJ, 14-Nov-2019.)
Assertion
Ref Expression
ax-infvn  |-  E. x
(Ind  x  /\  A. y (Ind  y  ->  x 
C_  y ) )
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 10906 . . 3  wff Ind  x
4 vy . . . . . . 7  setvar  y
54cv 1284 . . . . . 6  class  y
65wind 10906 . . . . 5  wff Ind  y
72, 5wss 2974 . . . . 5  wff  x  C_  y
86, 7wi 4 . . . 4  wff  (Ind  y  ->  x  C_  y
)
98, 4wal 1283 . . 3  wff  A. y
(Ind  y  ->  x  C_  y )
103, 9wa 102 . 2  wff  (Ind  x  /\  A. y (Ind  y  ->  x  C_  y
) )
1110, 1wex 1422 1  wff  E. x
(Ind  x  /\  A. y (Ind  y  ->  x 
C_  y ) )
Colors of variables: wff set class
This axiom is referenced by:  bj-omex  10922
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