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Theorem nobndlem7 29063
Description: Lemma for nobndup 29065 and nobnddown 29066. 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 29062 . 2  |-  ( ( F  C_  No  /\  A  e.  F )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  C
)
41elexi 3123 . . 3  |-  X  e. 
_V
54fvconst2 6116 . 2  |-  ( |^| { x  e.  On  | 
( A `  x
)  =/=  X }  e.  C  ->  ( ( C  X.  { X } ) `  |^| { x  e.  On  | 
( A `  x
)  =/=  X }
)  =  X )
63, 5syl 16 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 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818    C_ wss 3476   {csn 4027   {cpr 4029   |^|cint 4282   Oncon0 4878    X. cxp 4997   ` cfv 5588   1oc1o 7123   2oc2o 7124   Nocsur 29005
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 6576
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-1o 7130  df-2o 7131  df-no 29008  df-bday 29010
This theorem is referenced by:  nobndup  29065
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