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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 ι. (Contributed by Mario Carneiro, 10-Oct-2014.)
Assertion
Ref Expression
ax-inf ⊤⊧(λf:(ιι) [(1-1 f:(ιι)) (¬ (onto f:(ιι)))])

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