Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  axfelem1 Unicode version

Theorem axfelem1 23680
Description: Lemma for axfe (future) . 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
axfelem1  |-  ( A  e.  V  ->  suc  U. ( bday " A
)  e.  On )

Proof of Theorem axfelem1
StepHypRef Expression
1 bdayfun 23663 . . . . 5  |-  Fun  bday
2 funimaexg 5232 . . . . 5  |-  ( ( Fun  bday  /\  A  e.  V )  ->  ( bday " A )  e. 
_V )
31, 2mpan 654 . . . 4  |-  ( A  e.  V  ->  ( bday " A )  e. 
_V )
4 uniexg 4454 . . . 4  |-  ( (
bday " A )  e. 
_V  ->  U. ( bday " A
)  e.  _V )
53, 4syl 17 . . 3  |-  ( A  e.  V  ->  U. ( bday " A )  e. 
_V )
6 imassrn 4978 . . . . 5  |-  ( bday " A )  C_  ran  bday
7 bdayrn 23664 . . . . 5  |-  ran  bday  =  On
86, 7sseqtri 3152 . . . 4  |-  ( bday " A )  C_  On
9 ssorduni 4514 . . . 4  |-  ( (
bday " A )  C_  On  ->  Ord  U. ( bday " A ) )
108, 9ax-mp 10 . . 3  |-  Ord  U. ( bday " A )
115, 10jctil 525 . 2  |-  ( A  e.  V  ->  ( Ord  U. ( bday " A
)  /\  U. ( bday " A )  e. 
_V ) )
12 elon2 4340 . . 3  |-  ( U. ( bday " A )  e.  On  <->  ( Ord  U. ( bday " A
)  /\  U. ( bday " A )  e. 
_V ) )
13 sucelon 4545 . . 3  |-  ( U. ( bday " A )  e.  On  <->  suc  U. ( bday " A )  e.  On )
1412, 13bitr3i 244 . 2  |-  ( ( Ord  U. ( bday " A )  /\  U. ( bday " A )  e.  _V )  <->  suc  U. ( bday " A )  e.  On )
1511, 14sylib 190 1  |-  ( A  e.  V  ->  suc  U. ( bday " A
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1621   _Vcvv 2740    C_ wss 3094   U.cuni 3768   Ord word 4328   Oncon0 4329   suc csuc 4331   ran crn 4627   "cima 4629   Fun wfun 4632   bdaycbday 23630
This theorem is referenced by:  axfelem2  23681  axfelem14  23693
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-suc 4335  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-1o 6412  df-no 23631  df-bday 23633
  Copyright terms: Public domain W3C validator