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Theorem nobndlem1 29669
Description: Lemma for nobndup 29677 and nobnddown 29678. 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 29653 . . . . 5  |-  Fun  bday
2 funimaexg 5671 . . . . 5  |-  ( ( Fun  bday  /\  A  e.  V )  ->  ( bday " A )  e. 
_V )
31, 2mpan 670 . . . 4  |-  ( A  e.  V  ->  ( bday " A )  e. 
_V )
4 uniexg 6596 . . . 4  |-  ( (
bday " A )  e. 
_V  ->  U. ( bday " A
)  e.  _V )
53, 4syl 16 . . 3  |-  ( A  e.  V  ->  U. ( bday " A )  e. 
_V )
6 imassrn 5358 . . . . 5  |-  ( bday " A )  C_  ran  bday
7 bdayrn 29654 . . . . 5  |-  ran  bday  =  On
86, 7sseqtri 3531 . . . 4  |-  ( bday " A )  C_  On
9 ssorduni 6620 . . . 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 4898 . . 3  |-  ( U. ( bday " A )  e.  On  <->  ( Ord  U. ( bday " A
)  /\  U. ( bday " A )  e. 
_V ) )
13 sucelon 6651 . . 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 1819   _Vcvv 3109    C_ wss 3471   U.cuni 4251   Ord word 4886   Oncon0 4887   suc csuc 4889   ran crn 5009   "cima 5011   Fun wfun 5588   bdaycbday 29619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-1o 7148  df-no 29620  df-bday 29622
This theorem is referenced by:  nobndlem2  29670  nobndlem8  29676  nofulllem4  29682
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