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Theorem noxpsgn 24390
Description: The cross 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 5447 . . 3  |-  ( A  X.  { X }
) : A --> { 1o ,  2o }
3 feq2 5392 . . . 4  |-  ( x  =  A  ->  (
( A  X.  { X } ) : x --> { 1o ,  2o } 
<->  ( A  X.  { X } ) : A --> { 1o ,  2o }
) )
43rspcev 2897 . . 3  |-  ( ( A  e.  On  /\  ( A  X.  { X } ) : A --> { 1o ,  2o }
)  ->  E. x  e.  On  ( A  X.  { X } ) : x --> { 1o ,  2o } )
52, 4mpan2 652 . 2  |-  ( A  e.  On  ->  E. x  e.  On  ( A  X.  { X } ) : x --> { 1o ,  2o } )
6 elno 24371 . 2  |-  ( ( A  X.  { X } )  e.  No  <->  E. x  e.  On  ( A  X.  { X }
) : x --> { 1o ,  2o } )
75, 6sylibr 203 1  |-  ( A  e.  On  ->  ( A  X.  { X }
)  e.  No )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   E.wrex 2557   {csn 3653   {cpr 3654   Oncon0 4408    X. cxp 4703   -->wf 5267   1oc1o 6488   2oc2o 6489   Nocsur 24365
This theorem is referenced by:  noxp1o  24391  noxp2o  24392  nobndlem3  24419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-no 24368
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