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Theorem nobndlem5 29033
Description: Lemma for nobndup 29037 and nobnddown 29038. There is always a minimal value of a surreal that is not a given sign. (Contributed by Scott Fenton, 3-Aug-2011.)
Hypothesis
Ref Expression
nobndlem4.1  |-  X  e. 
{ 1o ,  2o }
Assertion
Ref Expression
nobndlem5  |-  ( A  e.  No  ->  ( A `  |^| { x  e.  On  |  ( A `
 x )  =/= 
X } )  =/= 
X )
Distinct variable groups:    x, A    x, X

Proof of Theorem nobndlem5
StepHypRef Expression
1 ssrab2 3585 . . . 4  |-  { x  e.  On  |  ( A `
 x )  =/= 
X }  C_  On
2 nobndlem4.1 . . . . . . . 8  |-  X  e. 
{ 1o ,  2o }
32nosgnn0i 28996 . . . . . . 7  |-  (/)  =/=  X
4 fvnobday 29019 . . . . . . . 8  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
54neeq1d 2744 . . . . . . 7  |-  ( A  e.  No  ->  (
( A `  ( bday `  A ) )  =/=  X  <->  (/)  =/=  X
) )
63, 5mpbiri 233 . . . . . 6  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =/= 
X )
7 bdayelon 29017 . . . . . . 7  |-  ( bday `  A )  e.  On
8 fveq2 5864 . . . . . . . . 9  |-  ( x  =  ( bday `  A
)  ->  ( A `  x )  =  ( A `  ( bday `  A ) ) )
98neeq1d 2744 . . . . . . . 8  |-  ( x  =  ( bday `  A
)  ->  ( ( A `  x )  =/=  X  <->  ( A `  ( bday `  A )
)  =/=  X ) )
109elrab3 3262 . . . . . . 7  |-  ( (
bday `  A )  e.  On  ->  ( ( bday `  A )  e. 
{ x  e.  On  |  ( A `  x )  =/=  X } 
<->  ( A `  ( bday `  A ) )  =/=  X ) )
117, 10ax-mp 5 . . . . . 6  |-  ( (
bday `  A )  e.  { x  e.  On  |  ( A `  x )  =/=  X } 
<->  ( A `  ( bday `  A ) )  =/=  X )
126, 11sylibr 212 . . . . 5  |-  ( A  e.  No  ->  ( bday `  A )  e. 
{ x  e.  On  |  ( A `  x )  =/=  X } )
13 ne0i 3791 . . . . 5  |-  ( (
bday `  A )  e.  { x  e.  On  |  ( A `  x )  =/=  X }  ->  { x  e.  On  |  ( A `
 x )  =/= 
X }  =/=  (/) )
1412, 13syl 16 . . . 4  |-  ( A  e.  No  ->  { x  e.  On  |  ( A `
 x )  =/= 
X }  =/=  (/) )
15 onint 6608 . . . 4  |-  ( ( { x  e.  On  |  ( A `  x )  =/=  X }  C_  On  /\  {
x  e.  On  | 
( A `  x
)  =/=  X }  =/=  (/) )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  {
x  e.  On  | 
( A `  x
)  =/=  X }
)
161, 14, 15sylancr 663 . . 3  |-  ( A  e.  No  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  {
x  e.  On  | 
( A `  x
)  =/=  X }
)
17 nfrab1 3042 . . . . 5  |-  F/_ x { x  e.  On  |  ( A `  x )  =/=  X }
1817nfint 4292 . . . 4  |-  F/_ x |^| { x  e.  On  |  ( A `  x )  =/=  X }
19 nfcv 2629 . . . 4  |-  F/_ x On
20 nfcv 2629 . . . . . 6  |-  F/_ x A
2120, 18nffv 5871 . . . . 5  |-  F/_ x
( A `  |^| { x  e.  On  | 
( A `  x
)  =/=  X }
)
22 nfcv 2629 . . . . 5  |-  F/_ x X
2321, 22nfne 2798 . . . 4  |-  F/ x
( A `  |^| { x  e.  On  | 
( A `  x
)  =/=  X }
)  =/=  X
24 fveq2 5864 . . . . 5  |-  ( x  =  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  ->  ( A `  x )  =  ( A `  |^| { x  e.  On  |  ( A `  x )  =/=  X } ) )
2524neeq1d 2744 . . . 4  |-  ( x  =  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  ->  (
( A `  x
)  =/=  X  <->  ( A `  |^| { x  e.  On  |  ( A `
 x )  =/= 
X } )  =/= 
X ) )
2618, 19, 23, 25elrabf 3259 . . 3  |-  ( |^| { x  e.  On  | 
( A `  x
)  =/=  X }  e.  { x  e.  On  |  ( A `  x )  =/=  X } 
<->  ( |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  On  /\  ( A `  |^| { x  e.  On  | 
( A `  x
)  =/=  X }
)  =/=  X ) )
2716, 26sylib 196 . 2  |-  ( A  e.  No  ->  ( |^| { x  e.  On  |  ( A `  x )  =/=  X }  e.  On  /\  ( A `  |^| { x  e.  On  |  ( A `
 x )  =/= 
X } )  =/= 
X ) )
2827simprd 463 1  |-  ( A  e.  No  ->  ( A `  |^| { x  e.  On  |  ( A `
 x )  =/= 
X } )  =/= 
X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818    C_ wss 3476   (/)c0 3785   {cpr 4029   |^|cint 4282   Oncon0 4878   ` cfv 5586   1oc1o 7120   2oc2o 7121   Nocsur 28977   bdaycbday 28979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-1o 7127  df-2o 7128  df-no 28980  df-bday 28982
This theorem is referenced by:  nobndup  29037  nobnddown  29038
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