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Theorem nobndlem7 30372
Description: Lemma for nobndup 30374 and nobnddown 30375. Calculate the value of  ( C  X.  { X } ) at the minimal ordinal whose value is different from  X. (Contributed by Scott Fenton, 3-Aug-2011.)
Hypotheses
Ref Expression
nobndlem7.1  |-  X  e. 
{ 1o ,  2o }
nobndlem7.2  |-  C  = 
|^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X }
Assertion
Ref Expression
nobndlem7  |-  ( ( F  C_  No  /\  A  e.  F )  ->  (
( C  X.  { X } ) `  |^| { x  e.  On  | 
( A `  x
)  =/=  X }
)  =  X )
Distinct variable groups:    A, a,
b, x    F, a,
b    X, a, b, x   
n, a, b, A   
n, F    n, X
Allowed substitution hints:    C( x, n, a, b)    F( x)

Proof of Theorem nobndlem7
StepHypRef Expression
1 nobndlem7.1 . . 3  |-  X  e. 
{ 1o ,  2o }
2 nobndlem7.2 . . 3  |-  C  = 
|^| { a  e.  On  |  A. n  e.  F  E. b  e.  a 
( n `  b
)  =/=  X }
31, 2nobndlem6 30371 . 2  |-  ( ( F  C_  No  /\  A  e.  F )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  C
)
41elexi 3097 . . 3  |-  X  e. 
_V
54fvconst2 6135 . 2  |-  ( |^| { x  e.  On  | 
( A `  x
)  =/=  X }  e.  C  ->  ( ( C  X.  { X } ) `  |^| { x  e.  On  | 
( A `  x
)  =/=  X }
)  =  X )
63, 5syl 17 1  |-  ( ( F  C_  No  /\  A  e.  F )  ->  (
( C  X.  { X } ) `  |^| { x  e.  On  | 
( A `  x
)  =/=  X }
)  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783   {crab 2786    C_ wss 3442   {csn 4002   {cpr 4004   |^|cint 4258    X. cxp 4852   Oncon0 5442   ` cfv 5601   1oc1o 7183   2oc2o 7184   Nocsur 30314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-1o 7190  df-2o 7191  df-no 30317  df-bday 30319
This theorem is referenced by:  nobndup  30374
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