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Theorem fvnobday 25637
Description: The value of a surreal at its birthday is  (/). (Shortened proof on 2012-Apr-14, SF) (Contributed by Scott Fenton, 14-Jun-2011.)
Assertion
Ref Expression
fvnobday  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )

Proof of Theorem fvnobday
StepHypRef Expression
1 bdayelon 25635 . . . 4  |-  ( bday `  A )  e.  On
21onirri 4688 . . 3  |-  -.  ( bday `  A )  e.  ( bday `  A
)
3 bdayval 25603 . . . 4  |-  ( A  e.  No  ->  ( bday `  A )  =  dom  A )
43eleq2d 2503 . . 3  |-  ( A  e.  No  ->  (
( bday `  A )  e.  ( bday `  A
)  <->  ( bday `  A
)  e.  dom  A
) )
52, 4mtbii 294 . 2  |-  ( A  e.  No  ->  -.  ( bday `  A )  e.  dom  A )
6 ndmfv 5755 . 2  |-  ( -.  ( bday `  A
)  e.  dom  A  ->  ( A `  ( bday `  A ) )  =  (/) )
75, 6syl 16 1  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725   (/)c0 3628   dom cdm 4878   ` cfv 5454   Nocsur 25595   bdaycbday 25597
This theorem is referenced by:  nodenselem3  25638  nodense  25644  nobndlem2  25648  nobndlem4  25650  nobndlem5  25651  nobndlem6  25652  nobndlem8  25654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-no 25598  df-bday 25600
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