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Theorem nobndlem5 29696
Description: Lemma for nobndup 29700 and nobnddown 29701. 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 3571 . . . 4  |-  { x  e.  On  |  ( A `
 x )  =/= 
X }  C_  On
2 nobndlem4.1 . . . . . . . 8  |-  X  e. 
{ 1o ,  2o }
32nosgnn0i 29659 . . . . . . 7  |-  (/)  =/=  X
4 fvnobday 29682 . . . . . . . 8  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
54neeq1d 2731 . . . . . . 7  |-  ( A  e.  No  ->  (
( A `  ( bday `  A ) )  =/=  X  <->  (/)  =/=  X
) )
63, 5mpbiri 233 . . . . . 6  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =/= 
X )
7 bdayelon 29680 . . . . . . 7  |-  ( bday `  A )  e.  On
8 fveq2 5848 . . . . . . . . 9  |-  ( x  =  ( bday `  A
)  ->  ( A `  x )  =  ( A `  ( bday `  A ) ) )
98neeq1d 2731 . . . . . . . 8  |-  ( x  =  ( bday `  A
)  ->  ( ( A `  x )  =/=  X  <->  ( A `  ( bday `  A )
)  =/=  X ) )
109elrab3 3255 . . . . . . 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 3789 . . . . 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 6603 . . . 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 661 . . 3  |-  ( A  e.  No  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  {
x  e.  On  | 
( A `  x
)  =/=  X }
)
17 nfrab1 3035 . . . . 5  |-  F/_ x { x  e.  On  |  ( A `  x )  =/=  X }
1817nfint 4281 . . . 4  |-  F/_ x |^| { x  e.  On  |  ( A `  x )  =/=  X }
19 nfcv 2616 . . . 4  |-  F/_ x On
20 nfcv 2616 . . . . . 6  |-  F/_ x A
2120, 18nffv 5855 . . . . 5  |-  F/_ x
( A `  |^| { x  e.  On  | 
( A `  x
)  =/=  X }
)
22 nfcv 2616 . . . . 5  |-  F/_ x X
2321, 22nfne 2785 . . . 4  |-  F/ x
( A `  |^| { x  e.  On  | 
( A `  x
)  =/=  X }
)  =/=  X
24 fveq2 5848 . . . . 5  |-  ( x  =  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  ->  ( A `  x )  =  ( A `  |^| { x  e.  On  |  ( A `  x )  =/=  X } ) )
2524neeq1d 2731 . . . 4  |-  ( x  =  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  ->  (
( A `  x
)  =/=  X  <->  ( A `  |^| { x  e.  On  |  ( A `
 x )  =/= 
X } )  =/= 
X ) )
2618, 19, 23, 25elrabf 3252 . . 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 461 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808    C_ wss 3461   (/)c0 3783   {cpr 4018   |^|cint 4271   Oncon0 4867   ` cfv 5570   1oc1o 7115   2oc2o 7116   Nocsur 29640   bdaycbday 29642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-1o 7122  df-2o 7123  df-no 29643  df-bday 29645
This theorem is referenced by:  nobndup  29700  nobnddown  29701
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