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Mirrors > Home > MPE Home > Th. List > alephon | Structured version Visualization version GIF version |
Description: An aleph is an ordinal number. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Ref | Expression |
---|---|
alephon | β’ (β΅βπ΄) β On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephfnon 10060 | . . 3 β’ β΅ Fn On | |
2 | fveq2 6892 | . . . . . 6 β’ (π₯ = β β (β΅βπ₯) = (β΅ββ )) | |
3 | 2 | eleq1d 2819 | . . . . 5 β’ (π₯ = β β ((β΅βπ₯) β On β (β΅ββ ) β On)) |
4 | fveq2 6892 | . . . . . 6 β’ (π₯ = π¦ β (β΅βπ₯) = (β΅βπ¦)) | |
5 | 4 | eleq1d 2819 | . . . . 5 β’ (π₯ = π¦ β ((β΅βπ₯) β On β (β΅βπ¦) β On)) |
6 | fveq2 6892 | . . . . . 6 β’ (π₯ = suc π¦ β (β΅βπ₯) = (β΅βsuc π¦)) | |
7 | 6 | eleq1d 2819 | . . . . 5 β’ (π₯ = suc π¦ β ((β΅βπ₯) β On β (β΅βsuc π¦) β On)) |
8 | aleph0 10061 | . . . . . 6 β’ (β΅ββ ) = Ο | |
9 | omelon 9641 | . . . . . 6 β’ Ο β On | |
10 | 8, 9 | eqeltri 2830 | . . . . 5 β’ (β΅ββ ) β On |
11 | alephsuc 10063 | . . . . . . 7 β’ (π¦ β On β (β΅βsuc π¦) = (harβ(β΅βπ¦))) | |
12 | harcl 9554 | . . . . . . 7 β’ (harβ(β΅βπ¦)) β On | |
13 | 11, 12 | eqeltrdi 2842 | . . . . . 6 β’ (π¦ β On β (β΅βsuc π¦) β On) |
14 | 13 | a1d 25 | . . . . 5 β’ (π¦ β On β ((β΅βπ¦) β On β (β΅βsuc π¦) β On)) |
15 | vex 3479 | . . . . . . 7 β’ π₯ β V | |
16 | iunon 8339 | . . . . . . 7 β’ ((π₯ β V β§ βπ¦ β π₯ (β΅βπ¦) β On) β βͺ π¦ β π₯ (β΅βπ¦) β On) | |
17 | 15, 16 | mpan 689 | . . . . . 6 β’ (βπ¦ β π₯ (β΅βπ¦) β On β βͺ π¦ β π₯ (β΅βπ¦) β On) |
18 | alephlim 10062 | . . . . . . . 8 β’ ((π₯ β V β§ Lim π₯) β (β΅βπ₯) = βͺ π¦ β π₯ (β΅βπ¦)) | |
19 | 15, 18 | mpan 689 | . . . . . . 7 β’ (Lim π₯ β (β΅βπ₯) = βͺ π¦ β π₯ (β΅βπ¦)) |
20 | 19 | eleq1d 2819 | . . . . . 6 β’ (Lim π₯ β ((β΅βπ₯) β On β βͺ π¦ β π₯ (β΅βπ¦) β On)) |
21 | 17, 20 | imbitrrid 245 | . . . . 5 β’ (Lim π₯ β (βπ¦ β π₯ (β΅βπ¦) β On β (β΅βπ₯) β On)) |
22 | 3, 5, 7, 5, 10, 14, 21 | tfinds 7849 | . . . 4 β’ (π¦ β On β (β΅βπ¦) β On) |
23 | 22 | rgen 3064 | . . 3 β’ βπ¦ β On (β΅βπ¦) β On |
24 | ffnfv 7118 | . . 3 β’ (β΅:OnβΆOn β (β΅ Fn On β§ βπ¦ β On (β΅βπ¦) β On)) | |
25 | 1, 23, 24 | mpbir2an 710 | . 2 β’ β΅:OnβΆOn |
26 | 0elon 6419 | . 2 β’ β β On | |
27 | 25, 26 | f0cli 7100 | 1 β’ (β΅βπ΄) β On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 βwral 3062 Vcvv 3475 β c0 4323 βͺ ciun 4998 Oncon0 6365 Lim wlim 6366 suc csuc 6367 Fn wfn 6539 βΆwf 6540 βcfv 6544 Οcom 7855 harchar 9551 β΅cale 9931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-en 8940 df-dom 8941 df-oi 9505 df-har 9552 df-aleph 9935 |
This theorem is referenced by: alephnbtwn 10066 alephnbtwn2 10067 alephordilem1 10068 alephord 10070 alephord2 10071 alephord3 10073 alephsucdom 10074 alephsuc2 10075 alephf1 10080 alephsdom 10081 alephdom2 10082 alephle 10083 cardaleph 10084 alephf1ALT 10098 alephfp 10103 dfac12k 10142 alephsing 10271 alephval2 10567 alephadd 10572 alephmul 10573 alephexp1 10574 alephsuc3 10575 alephreg 10577 pwcfsdom 10578 cfpwsdom 10579 gchaleph 10666 gchaleph2 10667 gch2 10670 minregex2 42286 alephiso2 42309 |
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