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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 10062 | . . 3 β’ β΅ Fn On | |
2 | fveq2 6890 | . . . . . 6 β’ (π₯ = β β (β΅βπ₯) = (β΅ββ )) | |
3 | 2 | eleq1d 2816 | . . . . 5 β’ (π₯ = β β ((β΅βπ₯) β On β (β΅ββ ) β On)) |
4 | fveq2 6890 | . . . . . 6 β’ (π₯ = π¦ β (β΅βπ₯) = (β΅βπ¦)) | |
5 | 4 | eleq1d 2816 | . . . . 5 β’ (π₯ = π¦ β ((β΅βπ₯) β On β (β΅βπ¦) β On)) |
6 | fveq2 6890 | . . . . . 6 β’ (π₯ = suc π¦ β (β΅βπ₯) = (β΅βsuc π¦)) | |
7 | 6 | eleq1d 2816 | . . . . 5 β’ (π₯ = suc π¦ β ((β΅βπ₯) β On β (β΅βsuc π¦) β On)) |
8 | aleph0 10063 | . . . . . 6 β’ (β΅ββ ) = Ο | |
9 | omelon 9643 | . . . . . 6 β’ Ο β On | |
10 | 8, 9 | eqeltri 2827 | . . . . 5 β’ (β΅ββ ) β On |
11 | alephsuc 10065 | . . . . . . 7 β’ (π¦ β On β (β΅βsuc π¦) = (harβ(β΅βπ¦))) | |
12 | harcl 9556 | . . . . . . 7 β’ (harβ(β΅βπ¦)) β On | |
13 | 11, 12 | eqeltrdi 2839 | . . . . . 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 3476 | . . . . . . 7 β’ π₯ β V | |
16 | iunon 8341 | . . . . . . 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 686 | . . . . . 6 β’ (βπ¦ β π₯ (β΅βπ¦) β On β βͺ π¦ β π₯ (β΅βπ¦) β On) |
18 | alephlim 10064 | . . . . . . . 8 β’ ((π₯ β V β§ Lim π₯) β (β΅βπ₯) = βͺ π¦ β π₯ (β΅βπ¦)) | |
19 | 15, 18 | mpan 686 | . . . . . . 7 β’ (Lim π₯ β (β΅βπ₯) = βͺ π¦ β π₯ (β΅βπ¦)) |
20 | 19 | eleq1d 2816 | . . . . . 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 7851 | . . . 4 β’ (π¦ β On β (β΅βπ¦) β On) |
23 | 22 | rgen 3061 | . . 3 β’ βπ¦ β On (β΅βπ¦) β On |
24 | ffnfv 7119 | . . 3 β’ (β΅:OnβΆOn β (β΅ Fn On β§ βπ¦ β On (β΅βπ¦) β On)) | |
25 | 1, 23, 24 | mpbir2an 707 | . 2 β’ β΅:OnβΆOn |
26 | 0elon 6417 | . 2 β’ β β On | |
27 | 25, 26 | f0cli 7098 | 1 β’ (β΅βπ΄) β On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 β wcel 2104 βwral 3059 Vcvv 3472 β c0 4321 βͺ ciun 4996 Oncon0 6363 Lim wlim 6364 suc csuc 6365 Fn wfn 6537 βΆwf 6538 βcfv 6542 Οcom 7857 harchar 9553 β΅cale 9933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-en 8942 df-dom 8943 df-oi 9507 df-har 9554 df-aleph 9937 |
This theorem is referenced by: alephnbtwn 10068 alephnbtwn2 10069 alephordilem1 10070 alephord 10072 alephord2 10073 alephord3 10075 alephsucdom 10076 alephsuc2 10077 alephf1 10082 alephsdom 10083 alephdom2 10084 alephle 10085 cardaleph 10086 alephf1ALT 10100 alephfp 10105 dfac12k 10144 alephsing 10273 alephval2 10569 alephadd 10574 alephmul 10575 alephexp1 10576 alephsuc3 10577 alephreg 10579 pwcfsdom 10580 cfpwsdom 10581 gchaleph 10668 gchaleph2 10669 gch2 10672 minregex2 42588 alephiso2 42611 |
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