Metamath Proof Explorer


Theorem isinf

Description: Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set hascountably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by Mario Carneiro, 15-Jan-2013) Avoid ax-pow . (Revised by BTernaryTau, 2-Jan-2025)

Ref Expression
Assertion isinf ¬ A Fin n ω x x A x n

Proof

Step Hyp Ref Expression
1 breq2 n = x n x
2 1 anbi2d n = x A x n x A x
3 2 exbidv n = x x A x n x x A x
4 breq2 n = m x n x m
5 4 anbi2d n = m x A x n x A x m
6 5 exbidv n = m x x A x n x x A x m
7 sseq1 x = y x A y A
8 7 adantl n = suc m x = y x A y A
9 breq1 x = y x n y n
10 breq2 n = suc m y n y suc m
11 9 10 sylan9bbr n = suc m x = y x n y suc m
12 8 11 anbi12d n = suc m x = y x A x n y A y suc m
13 12 cbvexdvaw n = suc m x x A x n y y A y suc m
14 0ss A
15 peano1 ω
16 enrefnn ω
17 15 16 ax-mp
18 0ex V
19 sseq1 x = x A A
20 breq1 x = x
21 19 20 anbi12d x = x A x A
22 18 21 spcev A x x A x
23 14 17 22 mp2an x x A x
24 23 a1i ¬ A Fin x x A x
25 ssdif0 A x A x =
26 eqss x = A x A A x
27 breq1 x = A x m A m
28 27 biimpa x = A x m A m
29 rspe m ω A m m ω A m
30 28 29 sylan2 m ω x = A x m m ω A m
31 isfi A Fin m ω A m
32 30 31 sylibr m ω x = A x m A Fin
33 32 expcom x = A x m m ω A Fin
34 26 33 sylanbr x A A x x m m ω A Fin
35 34 ex x A A x x m m ω A Fin
36 25 35 sylan2br x A A x = x m m ω A Fin
37 36 expcom A x = x A x m m ω A Fin
38 37 3impd A x = x A x m m ω A Fin
39 38 com12 x A x m m ω A x = A Fin
40 39 con3d x A x m m ω ¬ A Fin ¬ A x =
41 bren x m f f : x 1-1 onto m
42 neq0 ¬ A x = z z A x
43 eldifi z A x z A
44 43 snssd z A x z A
45 unss x A z A x z A
46 45 biimpi x A z A x z A
47 44 46 sylan2 x A z A x x z A
48 47 ad2ant2r x A f : x 1-1 onto m z A x m ω x z A
49 vex z V
50 vex m V
51 49 50 f1osn z m : z 1-1 onto m
52 51 jctr f : x 1-1 onto m f : x 1-1 onto m z m : z 1-1 onto m
53 eldifn z A x ¬ z x
54 disjsn x z = ¬ z x
55 53 54 sylibr z A x x z =
56 nnord m ω Ord m
57 orddisj Ord m m m =
58 56 57 syl m ω m m =
59 55 58 anim12i z A x m ω x z = m m =
60 f1oun f : x 1-1 onto m z m : z 1-1 onto m x z = m m = f z m : x z 1-1 onto m m
61 52 59 60 syl2an f : x 1-1 onto m z A x m ω f z m : x z 1-1 onto m m
62 df-suc suc m = m m
63 f1oeq3 suc m = m m f z m : x z 1-1 onto suc m f z m : x z 1-1 onto m m
64 62 63 ax-mp f z m : x z 1-1 onto suc m f z m : x z 1-1 onto m m
65 vex f V
66 snex z m V
67 65 66 unex f z m V
68 f1oeq1 g = f z m g : x z 1-1 onto suc m f z m : x z 1-1 onto suc m
69 67 68 spcev f z m : x z 1-1 onto suc m g g : x z 1-1 onto suc m
70 bren x z suc m g g : x z 1-1 onto suc m
71 69 70 sylibr f z m : x z 1-1 onto suc m x z suc m
72 64 71 sylbir f z m : x z 1-1 onto m m x z suc m
73 61 72 syl f : x 1-1 onto m z A x m ω x z suc m
74 73 adantll x A f : x 1-1 onto m z A x m ω x z suc m
75 vex x V
76 snex z V
77 75 76 unex x z V
78 sseq1 y = x z y A x z A
79 breq1 y = x z y suc m x z suc m
80 78 79 anbi12d y = x z y A y suc m x z A x z suc m
81 77 80 spcev x z A x z suc m y y A y suc m
82 48 74 81 syl2anc x A f : x 1-1 onto m z A x m ω y y A y suc m
83 82 expcom z A x m ω x A f : x 1-1 onto m y y A y suc m
84 83 ex z A x m ω x A f : x 1-1 onto m y y A y suc m
85 84 exlimiv z z A x m ω x A f : x 1-1 onto m y y A y suc m
86 42 85 sylbi ¬ A x = m ω x A f : x 1-1 onto m y y A y suc m
87 86 com13 x A f : x 1-1 onto m m ω ¬ A x = y y A y suc m
88 87 expcom f : x 1-1 onto m x A m ω ¬ A x = y y A y suc m
89 88 exlimiv f f : x 1-1 onto m x A m ω ¬ A x = y y A y suc m
90 41 89 sylbi x m x A m ω ¬ A x = y y A y suc m
91 90 3imp21 x A x m m ω ¬ A x = y y A y suc m
92 40 91 syld x A x m m ω ¬ A Fin y y A y suc m
93 92 3expia x A x m m ω ¬ A Fin y y A y suc m
94 93 exlimiv x x A x m m ω ¬ A Fin y y A y suc m
95 94 com3l m ω ¬ A Fin x x A x m y y A y suc m
96 3 6 13 24 95 finds2 n ω ¬ A Fin x x A x n
97 96 com12 ¬ A Fin n ω x x A x n
98 97 ralrimiv ¬ A Fin n ω x x A x n