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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 27565 | . . . . . 6 β’ bday : No βontoβOn | |
2 | fofun 6800 | . . . . . 6 β’ ( bday : No βontoβOn β Fun bday ) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 β’ Fun bday |
4 | funimaexg 6628 | . . . . 5 β’ ((Fun bday β§ π΄ β π) β ( bday β π΄) β V) | |
5 | 3, 4 | mpan 687 | . . . 4 β’ (π΄ β π β ( bday β π΄) β V) |
6 | 5 | uniexd 7729 | . . 3 β’ (π΄ β π β βͺ ( bday β π΄) β V) |
7 | imassrn 6064 | . . . . 5 β’ ( bday β π΄) β ran bday | |
8 | forn 6802 | . . . . . 6 β’ ( bday : No βontoβOn β ran bday = On) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 β’ ran bday = On |
10 | 7, 9 | sseqtri 4013 | . . . 4 β’ ( bday β π΄) β On |
11 | ssorduni 7763 | . . . 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 519 | . 2 β’ (π΄ β π β (Ord βͺ ( bday β π΄) β§ βͺ ( bday β π΄) β V)) |
14 | elon2 6369 | . . 3 β’ (βͺ ( bday β π΄) β On β (Ord βͺ ( bday β π΄) β§ βͺ ( bday β π΄) β V)) | |
15 | onsucb 7802 | . . 3 β’ (βͺ ( bday β π΄) β On β suc βͺ ( bday β π΄) β On) | |
16 | 14, 15 | bitr3i 277 | . 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 395 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 βͺ cuni 4902 ran crn 5670 β cima 5672 Ord word 6357 Oncon0 6358 suc csuc 6360 Fun wfun 6531 βontoβwfo 6535 No csur 27528 bday cbday 27530 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-1o 8467 df-no 27531 df-bday 27533 |
This theorem is referenced by: noetasuplem1 27621 noetainflem1 27625 |
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