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Theorem nobndlem7 27978
Description: Lemma for nobndup 27980 and nobnddown 27981. 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 27977 . 2  |-  ( ( F  C_  No  /\  A  e.  F )  ->  |^| { x  e.  On  |  ( A `
 x )  =/= 
X }  e.  C
)
41elexi 3082 . . 3  |-  X  e. 
_V
54fvconst2 6037 . 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 1370    e. wcel 1758    =/= wne 2645   A.wral 2796   E.wrex 2797   {crab 2800    C_ wss 3431   {csn 3980   {cpr 3982   |^|cint 4231   Oncon0 4822    X. cxp 4941   ` cfv 5521   1oc1o 7018   2oc2o 7019   Nocsur 27920
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-1o 7025  df-2o 7026  df-no 27923  df-bday 27925
This theorem is referenced by:  nobndup  27980
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