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Axiom ax-inf 202
Description: The axiom of infinity: the set of "individuals" is not Dedekind-finite. Using the axiom of choice, we can show that this is equivalent to an embedding of the natural numbers in iota. (Contributed by Mario Carneiro, 10-Oct-2014.)
Assertion
Ref Expression
ax-inf |- T. |= (E.\f:(iota -> iota) [(1-1 f:(iota -> iota)) /\ (~ (onto f:(iota -> iota)))])

Detailed syntax breakdown of Axiom ax-inf
StepHypRef Expression
1 kt 8 . 2 term T.
2 tex 123 . . 3 term E.
3 hi 4 . . . . 5 type iota
43, 3ht 2 . . . 4 type (iota -> iota)
5 vf . . . 4 var f
6 tf11 189 . . . . . 6 term 1-1
74, 5tv 1 . . . . . 6 term f:(iota -> iota)
86, 7kc 5 . . . . 5 term (1-1 f:(iota -> iota))
9 tne 120 . . . . . 6 term ~
10 tfo 190 . . . . . . 7 term onto
1110, 7kc 5 . . . . . 6 term (onto f:(iota -> iota))
129, 11kc 5 . . . . 5 term (~ (onto f:(iota -> iota)))
13 tan 119 . . . . 5 term /\
148, 12, 13kbr 9 . . . 4 term [(1-1 f:(iota -> iota)) /\ (~ (onto f:(iota -> iota)))]
154, 5, 14kl 6 . . 3 term \f:(iota -> iota) [(1-1 f:(iota -> iota)) /\ (~ (onto f:(iota -> iota)))]
162, 15kc 5 . 2 term (E.\f:(iota -> iota) [(1-1 f:(iota -> iota)) /\ (~ (onto f:(iota -> iota)))])
171, 16wffMMJ2 11 1 wff T. |= (E.\f:(iota -> iota) [(1-1 f:(iota -> iota)) /\ (~ (onto f:(iota -> iota)))])
Colors of variables: type var term
This axiom is referenced by: (None)
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