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Mirrors > Home > MPE Home > Th. List > bdayelon | Structured version Visualization version GIF version |
Description: The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by Scott Fenton, 8-Dec-2021.) |
Ref | Expression |
---|---|
bdayelon | β’ ( bday βπ΄) β On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdayfo 27041 | . . 3 β’ bday : No βontoβOn | |
2 | fof 6757 | . . 3 β’ ( bday : No βontoβOn β bday : No βΆOn) | |
3 | 1, 2 | ax-mp 5 | . 2 β’ bday : No βΆOn |
4 | 0elon 6372 | . 2 β’ β β On | |
5 | 3, 4 | f0cli 7049 | 1 β’ ( bday βπ΄) β On |
Colors of variables: wff setvar class |
Syntax hints: β wcel 2107 Oncon0 6318 βΆwf 6493 βontoβwfo 6495 βcfv 6497 No csur 27004 bday cbday 27006 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-1o 8413 df-no 27007 df-bday 27009 |
This theorem is referenced by: nocvxminlem 27139 scutbdaybnd2lim 27178 scutbdaylt 27179 slerec 27180 bday1s 27192 leftf 27217 rightf 27218 madebdayim 27239 oldbdayim 27240 oldirr 27241 madebdaylemold 27249 madebdaylemlrcut 27250 madebday 27251 newbday 27253 lrcut 27254 cofcutr 27265 lrrecval2 27274 lrrecpo 27275 addsproplem2 27304 addsproplem4 27306 addsproplem5 27307 addsproplem6 27308 addsproplem7 27309 addsprop 27310 negsproplem2 27349 negsproplem4 27351 negsproplem5 27352 negsproplem6 27353 negsproplem7 27354 negsprop 27355 mulsproplem2 27402 mulsproplem3 27403 mulsproplem4 27404 mulsproplem5 27405 mulsproplem6 27406 mulsproplem7 27407 mulsproplem8 27408 mulsproplem9 27409 |
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