Metamath Proof Explorer


Theorem minregex

Description: Given any cardinal number A , there exists an argument x , which yields the least regular uncountable value of aleph which is greater to or equal to A . This proof uses AC. (Contributed by RP, 23-Nov-2023)

Ref Expression
Assertion minregex A ran card ω x On x = y On | y A y cf y = y

Proof

Step Hyp Ref Expression
1 eldif A ran card ω A ran card ¬ A ω
2 omelon ω On
3 cardon card A On
4 eleq1 card A = A card A On A On
5 3 4 mpbii card A = A A On
6 ontri1 ω On A On ω A ¬ A ω
7 2 5 6 sylancr card A = A ω A ¬ A ω
8 7 pm5.32i card A = A ω A card A = A ¬ A ω
9 iscard4 card A = A A ran card
10 9 anbi1i card A = A ¬ A ω A ran card ¬ A ω
11 8 10 bitr2i A ran card ¬ A ω card A = A ω A
12 ancom card A = A ω A ω A card A = A
13 1 11 12 3bitri A ran card ω ω A card A = A
14 13 biimpi A ran card ω ω A card A = A
15 cardalephex ω A card A = A x On A = x
16 15 biimpa ω A card A = A x On A = x
17 eqimss A = x A x
18 17 a1i ω A card A = A A = x A x
19 18 reximdv ω A card A = A x On A = x x On A x
20 16 19 mpd ω A card A = A x On A x
21 onintrab2 x On A x x On | A x On
22 20 21 sylib ω A card A = A x On | A x On
23 simpr ω A card A = A x On | A x On x On | A x On
24 onsuc x On | A x On suc x On | A x On
25 23 24 syl ω A card A = A x On | A x On suc x On | A x On
26 eloni x On | A x On Ord x On | A x
27 23 26 syl ω A card A = A x On | A x On Ord x On | A x
28 0elsuc Ord x On | A x suc x On | A x
29 27 28 syl ω A card A = A x On | A x On suc x On | A x
30 cardaleph ω A card A = A A = x On | A x
31 30 adantr ω A card A = A x On | A x On A = x On | A x
32 sssucid x On | A x suc x On | A x
33 alephord3 x On | A x On suc x On | A x On x On | A x suc x On | A x x On | A x suc x On | A x
34 23 24 33 syl2anc2 ω A card A = A x On | A x On x On | A x suc x On | A x x On | A x suc x On | A x
35 32 34 mpbii ω A card A = A x On | A x On x On | A x suc x On | A x
36 31 35 eqsstrd ω A card A = A x On | A x On A suc x On | A x
37 alephreg cf suc x On | A x = suc x On | A x
38 37 a1i ω A card A = A x On | A x On cf suc x On | A x = suc x On | A x
39 29 36 38 3jca ω A card A = A x On | A x On suc x On | A x A suc x On | A x cf suc x On | A x = suc x On | A x
40 25 39 jca ω A card A = A x On | A x On suc x On | A x On suc x On | A x A suc x On | A x cf suc x On | A x = suc x On | A x
41 14 22 40 syl2anc2 A ran card ω suc x On | A x On suc x On | A x A suc x On | A x cf suc x On | A x = suc x On | A x
42 14 16 syl A ran card ω x On A = x
43 17 a1i A ran card ω A = x A x
44 43 reximdv A ran card ω x On A = x x On A x
45 42 44 mpd A ran card ω x On A x
46 45 21 sylib A ran card ω x On | A x On
47 46 24 syl A ran card ω suc x On | A x On
48 sbcan [˙ suc x On | A x / y]˙ y On y A y cf y = y [˙ suc x On | A x / y]˙ y On [˙ suc x On | A x / y]˙ y A y cf y = y
49 sbcel1v [˙ suc x On | A x / y]˙ y On suc x On | A x On
50 49 a1i suc x On | A x On [˙ suc x On | A x / y]˙ y On suc x On | A x On
51 sbc3an [˙ suc x On | A x / y]˙ y A y cf y = y [˙ suc x On | A x / y]˙ y [˙ suc x On | A x / y]˙ A y [˙ suc x On | A x / y]˙ cf y = y
52 sbcel2gv suc x On | A x On [˙ suc x On | A x / y]˙ y suc x On | A x
53 sbcssg suc x On | A x On [˙ suc x On | A x / y]˙ A y suc x On | A x / y A suc x On | A x / y y
54 csbconstg suc x On | A x On suc x On | A x / y A = A
55 csbfv2g suc x On | A x On suc x On | A x / y y = suc x On | A x / y y
56 csbvarg suc x On | A x On suc x On | A x / y y = suc x On | A x
57 56 fveq2d suc x On | A x On suc x On | A x / y y = suc x On | A x
58 55 57 eqtrd suc x On | A x On suc x On | A x / y y = suc x On | A x
59 54 58 sseq12d suc x On | A x On suc x On | A x / y A suc x On | A x / y y A suc x On | A x
60 53 59 bitrd suc x On | A x On [˙ suc x On | A x / y]˙ A y A suc x On | A x
61 sbceqg suc x On | A x On [˙ suc x On | A x / y]˙ cf y = y suc x On | A x / y cf y = suc x On | A x / y y
62 csbfv2g suc x On | A x On suc x On | A x / y cf y = cf suc x On | A x / y y
63 58 fveq2d suc x On | A x On cf suc x On | A x / y y = cf suc x On | A x
64 62 63 eqtrd suc x On | A x On suc x On | A x / y cf y = cf suc x On | A x
65 64 58 eqeq12d suc x On | A x On suc x On | A x / y cf y = suc x On | A x / y y cf suc x On | A x = suc x On | A x
66 61 65 bitrd suc x On | A x On [˙ suc x On | A x / y]˙ cf y = y cf suc x On | A x = suc x On | A x
67 52 60 66 3anbi123d suc x On | A x On [˙ suc x On | A x / y]˙ y [˙ suc x On | A x / y]˙ A y [˙ suc x On | A x / y]˙ cf y = y suc x On | A x A suc x On | A x cf suc x On | A x = suc x On | A x
68 51 67 bitrid suc x On | A x On [˙ suc x On | A x / y]˙ y A y cf y = y suc x On | A x A suc x On | A x cf suc x On | A x = suc x On | A x
69 50 68 anbi12d suc x On | A x On [˙ suc x On | A x / y]˙ y On [˙ suc x On | A x / y]˙ y A y cf y = y suc x On | A x On suc x On | A x A suc x On | A x cf suc x On | A x = suc x On | A x
70 48 69 bitrid suc x On | A x On [˙ suc x On | A x / y]˙ y On y A y cf y = y suc x On | A x On suc x On | A x A suc x On | A x cf suc x On | A x = suc x On | A x
71 47 70 syl A ran card ω [˙ suc x On | A x / y]˙ y On y A y cf y = y suc x On | A x On suc x On | A x A suc x On | A x cf suc x On | A x = suc x On | A x
72 41 71 mpbird A ran card ω [˙ suc x On | A x / y]˙ y On y A y cf y = y
73 72 spesbcd A ran card ω y y On y A y cf y = y
74 onintrab2 y On y A y cf y = y y On | y A y cf y = y On
75 df-rex y On y A y cf y = y y y On y A y cf y = y
76 risset y On | y A y cf y = y On x On x = y On | y A y cf y = y
77 74 75 76 3bitr3i y y On y A y cf y = y x On x = y On | y A y cf y = y
78 73 77 sylib A ran card ω x On x = y On | y A y cf y = y