|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 | ⊢ ( bday ‘𝐴) ∈ On | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bdayfo 27723 | . . 3 ⊢ bday : No –onto→On | |
| 2 | fof 6819 | . . 3 ⊢ ( bday : No –onto→On → bday : No ⟶On) | |
| 3 | 1, 2 | ax-mp 5 | . 2 ⊢ bday : No ⟶On | 
| 4 | 0elon 6437 | . 2 ⊢ ∅ ∈ On | |
| 5 | 3, 4 | f0cli 7117 | 1 ⊢ ( bday ‘𝐴) ∈ On | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 Oncon0 6383 ⟶wf 6556 –onto→wfo 6558 ‘cfv 6560 No csur 27685 bday cbday 27687 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-1o 8507 df-no 27688 df-bday 27690 | 
| This theorem is referenced by: nocvxminlem 27823 scutbdaybnd2lim 27863 scutbdaylt 27864 slerec 27865 bday1s 27877 cuteq1 27879 leftf 27905 rightf 27906 madebdayim 27927 oldbdayim 27928 oldirr 27929 madebdaylemold 27937 madebdaylemlrcut 27938 madebday 27939 newbday 27941 lrcut 27942 0elold 27948 cofcutr 27959 lrrecval2 27974 lrrecpo 27975 addsproplem2 28004 addsproplem4 28006 addsproplem5 28007 addsproplem6 28008 addsproplem7 28009 addsprop 28010 addsbdaylem 28050 addsbday 28051 negsproplem2 28062 negsproplem4 28064 negsproplem5 28065 negsproplem6 28066 negsproplem7 28067 negsprop 28068 negsbdaylem 28089 mulsproplem2 28144 mulsproplem3 28145 mulsproplem4 28146 mulsproplem5 28147 mulsproplem6 28148 mulsproplem7 28149 mulsproplem8 28150 mulsproplem12 28154 mulsproplem13 28155 mulsproplem14 28156 mulsprop 28157 sltonold 28284 n0sbday 28355 pw2bday 28419 zs12bday 28425 | 
| Copyright terms: Public domain | W3C validator |