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Theorem noxpsgn 30125
Description: The Cartesian product of an ordinal and the singleton of a sign is a surreal. (Contributed by Scott Fenton, 21-Jun-2011.)
Hypothesis
Ref Expression
noxpsgn.1  |-  X  e. 
{ 1o ,  2o }
Assertion
Ref Expression
noxpsgn  |-  ( A  e.  On  ->  ( A  X.  { X }
)  e.  No )

Proof of Theorem noxpsgn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 noxpsgn.1 . . . 4  |-  X  e. 
{ 1o ,  2o }
21fconst6 5758 . . 3  |-  ( A  X.  { X }
) : A --> { 1o ,  2o }
3 feq2 5697 . . . 4  |-  ( x  =  A  ->  (
( A  X.  { X } ) : x --> { 1o ,  2o } 
<->  ( A  X.  { X } ) : A --> { 1o ,  2o }
) )
43rspcev 3160 . . 3  |-  ( ( A  e.  On  /\  ( A  X.  { X } ) : A --> { 1o ,  2o }
)  ->  E. x  e.  On  ( A  X.  { X } ) : x --> { 1o ,  2o } )
52, 4mpan2 669 . 2  |-  ( A  e.  On  ->  E. x  e.  On  ( A  X.  { X } ) : x --> { 1o ,  2o } )
6 elno 30106 . 2  |-  ( ( A  X.  { X } )  e.  No  <->  E. x  e.  On  ( A  X.  { X }
) : x --> { 1o ,  2o } )
75, 6sylibr 212 1  |-  ( A  e.  On  ->  ( A  X.  { X }
)  e.  No )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842   E.wrex 2755   {csn 3972   {cpr 3974    X. cxp 4821   Oncon0 5410   -->wf 5565   1oc1o 7160   2oc2o 7161   Nocsur 30100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-no 30103
This theorem is referenced by:  noxp1o  30126  noxp2o  30127  nobndlem3  30154
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