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Theorem nobndlem1 25363
Description: Lemma for nobndup 25371 and nobnddown 25372. 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.)
Assertion
Ref Expression
nobndlem1  |-  ( A  e.  V  ->  suc  U. ( bday " A
)  e.  On )

Proof of Theorem nobndlem1
StepHypRef Expression
1 bdayfun 25347 . . . . 5  |-  Fun  bday
2 funimaexg 5463 . . . . 5  |-  ( ( Fun  bday  /\  A  e.  V )  ->  ( bday " A )  e. 
_V )
31, 2mpan 652 . . . 4  |-  ( A  e.  V  ->  ( bday " A )  e. 
_V )
4 uniexg 4639 . . . 4  |-  ( (
bday " A )  e. 
_V  ->  U. ( bday " A
)  e.  _V )
53, 4syl 16 . . 3  |-  ( A  e.  V  ->  U. ( bday " A )  e. 
_V )
6 imassrn 5149 . . . . 5  |-  ( bday " A )  C_  ran  bday
7 bdayrn 25348 . . . . 5  |-  ran  bday  =  On
86, 7sseqtri 3316 . . . 4  |-  ( bday " A )  C_  On
9 ssorduni 4699 . . . 4  |-  ( (
bday " A )  C_  On  ->  Ord  U. ( bday " A ) )
108, 9ax-mp 8 . . 3  |-  Ord  U. ( bday " A )
115, 10jctil 524 . 2  |-  ( A  e.  V  ->  ( Ord  U. ( bday " A
)  /\  U. ( bday " A )  e. 
_V ) )
12 elon2 4526 . . 3  |-  ( U. ( bday " A )  e.  On  <->  ( Ord  U. ( bday " A
)  /\  U. ( bday " A )  e. 
_V ) )
13 sucelon 4730 . . 3  |-  ( U. ( bday " A )  e.  On  <->  suc  U. ( bday " A )  e.  On )
1412, 13bitr3i 243 . 2  |-  ( ( Ord  U. ( bday " A )  /\  U. ( bday " A )  e.  _V )  <->  suc  U. ( bday " A )  e.  On )
1511, 14sylib 189 1  |-  ( A  e.  V  ->  suc  U. ( bday " A
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   _Vcvv 2892    C_ wss 3256   U.cuni 3950   Ord word 4514   Oncon0 4515   suc csuc 4517   ran crn 4812   "cima 4814   Fun wfun 5381   bdaycbday 25313
This theorem is referenced by:  nobndlem2  25364  nobndlem8  25370  nofulllem4  25376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-suc 4521  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-1o 6653  df-no 25314  df-bday 25316
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