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Theorem nobndlem1 27852
Description: Lemma for nobndup 27860 and nobnddown 27861. 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 27836 . . . . 5  |-  Fun  bday
2 funimaexg 5514 . . . . 5  |-  ( ( Fun  bday  /\  A  e.  V )  ->  ( bday " A )  e. 
_V )
31, 2mpan 670 . . . 4  |-  ( A  e.  V  ->  ( bday " A )  e. 
_V )
4 uniexg 6396 . . . 4  |-  ( (
bday " A )  e. 
_V  ->  U. ( bday " A
)  e.  _V )
53, 4syl 16 . . 3  |-  ( A  e.  V  ->  U. ( bday " A )  e. 
_V )
6 imassrn 5199 . . . . 5  |-  ( bday " A )  C_  ran  bday
7 bdayrn 27837 . . . . 5  |-  ran  bday  =  On
86, 7sseqtri 3407 . . . 4  |-  ( bday " A )  C_  On
9 ssorduni 6416 . . . 4  |-  ( (
bday " A )  C_  On  ->  Ord  U. ( bday " A ) )
108, 9ax-mp 5 . . 3  |-  Ord  U. ( bday " A )
115, 10jctil 537 . 2  |-  ( A  e.  V  ->  ( Ord  U. ( bday " A
)  /\  U. ( bday " A )  e. 
_V ) )
12 elon2 4749 . . 3  |-  ( U. ( bday " A )  e.  On  <->  ( Ord  U. ( bday " A
)  /\  U. ( bday " A )  e. 
_V ) )
13 sucelon 6447 . . 3  |-  ( U. ( bday " A )  e.  On  <->  suc  U. ( bday " A )  e.  On )
1412, 13bitr3i 251 . 2  |-  ( ( Ord  U. ( bday " A )  /\  U. ( bday " A )  e.  _V )  <->  suc  U. ( bday " A )  e.  On )
1511, 14sylib 196 1  |-  ( A  e.  V  ->  suc  U. ( bday " A
)  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   _Vcvv 2991    C_ wss 3347   U.cuni 4110   Ord word 4737   Oncon0 4738   suc csuc 4740   ran crn 4860   "cima 4862   Fun wfun 5431   bdaycbday 27802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-1o 6939  df-no 27803  df-bday 27805
This theorem is referenced by:  nobndlem2  27853  nobndlem8  27859  nofulllem4  27865
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