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Theorem nobndlem5 27957
Description: Lemma for nobndup 27961 and nobnddown 27962. 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 3521 . . . 4  |-  { x  e.  On  |  ( A `
 x )  =/= 
X }  C_  On
2 nobndlem4.1 . . . . . . . 8  |-  X  e. 
{ 1o ,  2o }
32nosgnn0i 27920 . . . . . . 7  |-  (/)  =/=  X
4 fvnobday 27943 . . . . . . . 8  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =  (/) )
54neeq1d 2722 . . . . . . 7  |-  ( A  e.  No  ->  (
( A `  ( bday `  A ) )  =/=  X  <->  (/)  =/=  X
) )
63, 5mpbiri 233 . . . . . 6  |-  ( A  e.  No  ->  ( A `  ( bday `  A ) )  =/= 
X )
7 bdayelon 27941 . . . . . . 7  |-  ( bday `  A )  e.  On
8 fveq2 5775 . . . . . . . . 9  |-  ( x  =  ( bday `  A
)  ->  ( A `  x )  =  ( A `  ( bday `  A ) ) )
98neeq1d 2722 . . . . . . . 8  |-  ( x  =  ( bday `  A
)  ->  ( ( A `  x )  =/=  X  <->  ( A `  ( bday `  A )
)  =/=  X ) )
109elrab3 3201 . . . . . . 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 3727 . . . . 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 6492 . . . 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 2983 . . . . 5  |-  F/_ x { x  e.  On  |  ( A `  x )  =/=  X }
1817nfint 4222 . . . 4  |-  F/_ x |^| { x  e.  On  |  ( A `  x )  =/=  X }
19 nfcv 2610 . . . 4  |-  F/_ x On
20 nfcv 2610 . . . . . 6  |-  F/_ x A
2120, 18nffv 5782 . . . . 5  |-  F/_ x
( A `  |^| { x  e.  On  | 
( A `  x
)  =/=  X }
)
22 nfcv 2610 . . . . 5  |-  F/_ x X
2321, 22nfne 2776 . . . 4  |-  F/ x
( A `  |^| { x  e.  On  | 
( A `  x
)  =/=  X }
)  =/=  X
24 fveq2 5775 . . . . 5  |-  ( x  =  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  ->  ( A `  x )  =  ( A `  |^| { x  e.  On  |  ( A `  x )  =/=  X } ) )
2524neeq1d 2722 . . . 4  |-  ( x  =  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  ->  (
( A `  x
)  =/=  X  <->  ( A `  |^| { x  e.  On  |  ( A `
 x )  =/= 
X } )  =/= 
X ) )
2618, 19, 23, 25elrabf 3198 . . 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 1370    e. wcel 1757    =/= wne 2641   {crab 2796    C_ wss 3412   (/)c0 3721   {cpr 3963   |^|cint 4212   Oncon0 4803   ` cfv 5502   1oc1o 6999   2oc2o 7000   Nocsur 27901   bdaycbday 27903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-1o 7006  df-2o 7007  df-no 27904  df-bday 27906
This theorem is referenced by:  nobndup  27961  nobnddown  27962
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