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Mirrors > Home > MPE Home > Th. List > cardaleph | Structured version Visualization version Unicode version |
Description: Given any transfinite cardinal number , there is exactly one aleph that is equal to it. Here we compute that aleph explicitly. (Contributed by NM, 9-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.) |
Ref | Expression |
---|---|
cardaleph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardon 8770 | . . . . . . . . 9 | |
2 | eleq1 2689 | . . . . . . . . 9 | |
3 | 1, 2 | mpbii 223 | . . . . . . . 8 |
4 | alephle 8911 | . . . . . . . . 9 | |
5 | fveq2 6191 | . . . . . . . . . . 11 | |
6 | 5 | sseq2d 3633 | . . . . . . . . . 10 |
7 | 6 | rspcev 3309 | . . . . . . . . 9 |
8 | 4, 7 | mpdan 702 | . . . . . . . 8 |
9 | nfcv 2764 | . . . . . . . . . 10 | |
10 | nfcv 2764 | . . . . . . . . . . 11 | |
11 | nfrab1 3122 | . . . . . . . . . . . 12 | |
12 | 11 | nfint 4486 | . . . . . . . . . . 11 |
13 | 10, 12 | nffv 6198 | . . . . . . . . . 10 |
14 | 9, 13 | nfss 3596 | . . . . . . . . 9 |
15 | fveq2 6191 | . . . . . . . . . 10 | |
16 | 15 | sseq2d 3633 | . . . . . . . . 9 |
17 | 14, 16 | onminsb 6999 | . . . . . . . 8 |
18 | 3, 8, 17 | 3syl 18 | . . . . . . 7 |
19 | 18 | a1i 11 | . . . . . 6 |
20 | fveq2 6191 | . . . . . . . . 9 | |
21 | aleph0 8889 | . . . . . . . . 9 | |
22 | 20, 21 | syl6eq 2672 | . . . . . . . 8 |
23 | 22 | sseq1d 3632 | . . . . . . 7 |
24 | 23 | biimprd 238 | . . . . . 6 |
25 | 19, 24 | anim12d 586 | . . . . 5 |
26 | eqss 3618 | . . . . 5 | |
27 | 25, 26 | syl6ibr 242 | . . . 4 |
28 | 27 | com12 32 | . . 3 |
29 | 28 | ancoms 469 | . 2 |
30 | vex 3203 | . . . . . . . . . . . 12 | |
31 | 30 | sucid 5804 | . . . . . . . . . . 11 |
32 | eleq2 2690 | . . . . . . . . . . 11 | |
33 | 31, 32 | mpbiri 248 | . . . . . . . . . 10 |
34 | fveq2 6191 | . . . . . . . . . . . 12 | |
35 | 34 | sseq2d 3633 | . . . . . . . . . . 11 |
36 | 35 | onnminsb 7004 | . . . . . . . . . 10 |
37 | 33, 36 | syl5 34 | . . . . . . . . 9 |
38 | 37 | imp 445 | . . . . . . . 8 |
39 | 38 | adantl 482 | . . . . . . 7 |
40 | fveq2 6191 | . . . . . . . . . . 11 | |
41 | alephsuc 8891 | . . . . . . . . . . 11 har | |
42 | 40, 41 | sylan9eqr 2678 | . . . . . . . . . 10 har |
43 | 42 | eleq2d 2687 | . . . . . . . . 9 har |
44 | 43 | biimpd 219 | . . . . . . . 8 har |
45 | elharval 8468 | . . . . . . . . . 10 har | |
46 | 45 | simprbi 480 | . . . . . . . . 9 har |
47 | onenon 8775 | . . . . . . . . . . . 12 | |
48 | 3, 47 | syl 17 | . . . . . . . . . . 11 |
49 | alephon 8892 | . . . . . . . . . . . 12 | |
50 | onenon 8775 | . . . . . . . . . . . 12 | |
51 | 49, 50 | ax-mp 5 | . . . . . . . . . . 11 |
52 | carddom2 8803 | . . . . . . . . . . 11 | |
53 | 48, 51, 52 | sylancl 694 | . . . . . . . . . 10 |
54 | sseq1 3626 | . . . . . . . . . . 11 | |
55 | alephcard 8893 | . . . . . . . . . . . 12 | |
56 | 55 | sseq2i 3630 | . . . . . . . . . . 11 |
57 | 54, 56 | syl6bb 276 | . . . . . . . . . 10 |
58 | 53, 57 | bitr3d 270 | . . . . . . . . 9 |
59 | 46, 58 | syl5ib 234 | . . . . . . . 8 har |
60 | 44, 59 | sylan9r 690 | . . . . . . 7 |
61 | 39, 60 | mtod 189 | . . . . . 6 |
62 | 61 | rexlimdvaa 3032 | . . . . 5 |
63 | onintrab2 7002 | . . . . . . . . . . . . . 14 | |
64 | 8, 63 | sylib 208 | . . . . . . . . . . . . 13 |
65 | onelon 5748 | . . . . . . . . . . . . 13 | |
66 | 64, 65 | sylan 488 | . . . . . . . . . . . 12 |
67 | 36 | adantld 483 | . . . . . . . . . . . 12 |
68 | 66, 67 | mpcom 38 | . . . . . . . . . . 11 |
69 | 49 | onelssi 5836 | . . . . . . . . . . 11 |
70 | 68, 69 | nsyl 135 | . . . . . . . . . 10 |
71 | 70 | nrexdv 3001 | . . . . . . . . 9 |
72 | 71 | adantr 481 | . . . . . . . 8 |
73 | alephlim 8890 | . . . . . . . . . . 11 | |
74 | 64, 73 | sylan 488 | . . . . . . . . . 10 |
75 | 74 | eleq2d 2687 | . . . . . . . . 9 |
76 | eliun 4524 | . . . . . . . . 9 | |
77 | 75, 76 | syl6bb 276 | . . . . . . . 8 |
78 | 72, 77 | mtbird 315 | . . . . . . 7 |
79 | 78 | ex 450 | . . . . . 6 |
80 | 3, 79 | syl 17 | . . . . 5 |
81 | 62, 80 | jaod 395 | . . . 4 |
82 | 8, 17 | syl 17 | . . . . . 6 |
83 | alephon 8892 | . . . . . . 7 | |
84 | onsseleq 5765 | . . . . . . 7 | |
85 | 83, 84 | mpan2 707 | . . . . . 6 |
86 | 82, 85 | mpbid 222 | . . . . 5 |
87 | 86 | ord 392 | . . . 4 |
88 | 3, 81, 87 | sylsyld 61 | . . 3 |
89 | 88 | adantl 482 | . 2 |
90 | eloni 5733 | . . . . 5 | |
91 | ordzsl 7045 | . . . . . 6 | |
92 | 3orass 1040 | . . . . . 6 | |
93 | 91, 92 | bitri 264 | . . . . 5 |
94 | 90, 93 | sylib 208 | . . . 4 |
95 | 3, 64, 94 | 3syl 18 | . . 3 |
96 | 95 | adantl 482 | . 2 |
97 | 29, 89, 96 | mpjaod 396 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 w3o 1036 wceq 1483 wcel 1990 wrex 2913 crab 2916 wss 3574 c0 3915 cint 4475 ciun 4520 class class class wbr 4653 cdm 5114 word 5722 con0 5723 wlim 5724 csuc 5725 cfv 5888 com 7065 cdom 7953 harchar 8461 ccrd 8761 cale 8762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-oi 8415 df-har 8463 df-card 8765 df-aleph 8766 |
This theorem is referenced by: cardalephex 8913 tskcard 9603 |
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