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Theorem axfelem1 23715
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 23698 . . . . 5  |-  Fun  bday
2 funimaexg 5267 . . . . 5  |-  ( ( Fun  bday  /\  A  e.  V )  ->  ( bday " A )  e. 
_V )
31, 2mpan 654 . . . 4  |-  ( A  e.  V  ->  ( bday " A )  e. 
_V )
4 uniexg 4489 . . . 4  |-  ( (
bday " A )  e. 
_V  ->  U. ( bday " A
)  e.  _V )
53, 4syl 17 . . 3  |-  ( A  e.  V  ->  U. ( bday " A )  e. 
_V )
6 imassrn 5013 . . . . 5  |-  ( bday " A )  C_  ran  bday
7 bdayrn 23699 . . . . 5  |-  ran  bday  =  On
86, 7sseqtri 3185 . . . 4  |-  ( bday " A )  C_  On
9 ssorduni 4549 . . . 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 4375 . . 3  |-  ( U. ( bday " A )  e.  On  <->  ( Ord  U. ( bday " A
)  /\  U. ( bday " A )  e. 
_V ) )
13 sucelon 4580 . . 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 2763    C_ wss 3127   U.cuni 3801   Ord word 4363   Oncon0 4364   suc csuc 4366   ran crn 4662   "cima 4664   Fun wfun 4667   bdaycbday 23665
This theorem is referenced by:  axfelem2  23716  axfelem14  23728
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-suc 4370  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-1o 6447  df-no 23666  df-bday 23668
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