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Mirrors > Home > MPE Home > Th. List > bdayimaon | Structured version Visualization version GIF version |
Description: Lemma for full-eta properties. The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.) |
Ref | Expression |
---|---|
bdayimaon | β’ (π΄ β π β suc βͺ ( bday β π΄) β On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdayfo 27638 | . . . . . 6 β’ bday : No βontoβOn | |
2 | fofun 6817 | . . . . . 6 β’ ( bday : No βontoβOn β Fun bday ) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 β’ Fun bday |
4 | funimaexg 6644 | . . . . 5 β’ ((Fun bday β§ π΄ β π) β ( bday β π΄) β V) | |
5 | 3, 4 | mpan 688 | . . . 4 β’ (π΄ β π β ( bday β π΄) β V) |
6 | 5 | uniexd 7755 | . . 3 β’ (π΄ β π β βͺ ( bday β π΄) β V) |
7 | imassrn 6079 | . . . . 5 β’ ( bday β π΄) β ran bday | |
8 | forn 6819 | . . . . . 6 β’ ( bday : No βontoβOn β ran bday = On) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 β’ ran bday = On |
10 | 7, 9 | sseqtri 4018 | . . . 4 β’ ( bday β π΄) β On |
11 | ssorduni 7789 | . . . 4 β’ (( bday β π΄) β On β Ord βͺ ( bday β π΄)) | |
12 | 10, 11 | ax-mp 5 | . . 3 β’ Ord βͺ ( bday β π΄) |
13 | 6, 12 | jctil 518 | . 2 β’ (π΄ β π β (Ord βͺ ( bday β π΄) β§ βͺ ( bday β π΄) β V)) |
14 | elon2 6385 | . . 3 β’ (βͺ ( bday β π΄) β On β (Ord βͺ ( bday β π΄) β§ βͺ ( bday β π΄) β V)) | |
15 | onsucb 7828 | . . 3 β’ (βͺ ( bday β π΄) β On β suc βͺ ( bday β π΄) β On) | |
16 | 14, 15 | bitr3i 276 | . 2 β’ ((Ord βͺ ( bday β π΄) β§ βͺ ( bday β π΄) β V) β suc βͺ ( bday β π΄) β On) |
17 | 13, 16 | sylib 217 | 1 β’ (π΄ β π β suc βͺ ( bday β π΄) β On) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3473 β wss 3949 βͺ cuni 4912 ran crn 5683 β cima 5685 Ord word 6373 Oncon0 6374 suc csuc 6376 Fun wfun 6547 βontoβwfo 6551 No csur 27601 bday cbday 27603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-1o 8495 df-no 27604 df-bday 27606 |
This theorem is referenced by: noetasuplem1 27694 noetainflem1 27698 |
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