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Theorem isf32lem3 8747
Description: Lemma for isfin3-2 8759. Being a chain, difference sets are disjoint (one case). (Contributed by Stefan O'Rear, 5-Nov-2014.)
Hypotheses
Ref Expression
isf32lem.a  |-  ( ph  ->  F : om --> ~P G
)
isf32lem.b  |-  ( ph  ->  A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x ) )
isf32lem.c  |-  ( ph  ->  -.  |^| ran  F  e. 
ran  F )
Assertion
Ref Expression
isf32lem3  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( ( ( F `
 A )  \ 
( F `  suc  A ) )  i^i  (
( F `  B
)  \  ( F `  suc  B ) ) )  =  (/) )
Distinct variable groups:    x, B    ph, x    x, A    x, F
Allowed substitution hint:    G( x)

Proof of Theorem isf32lem3
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eldifi 3631 . . . 4  |-  ( a  e.  ( ( F `
 A )  \ 
( F `  suc  A ) )  ->  a  e.  ( F `  A
) )
2 simpll 753 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  A  e.  om )
3 peano2 6715 . . . . . . 7  |-  ( B  e.  om  ->  suc  B  e.  om )
43ad2antlr 726 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  suc  B  e.  om )
5 nnord 6703 . . . . . . . 8  |-  ( A  e.  om  ->  Ord  A )
65ad2antrr 725 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  Ord  A )
7 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  B  e.  A )
8 ordsucss 6648 . . . . . . 7  |-  ( Ord 
A  ->  ( B  e.  A  ->  suc  B  C_  A ) )
96, 7, 8sylc 60 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  suc  B  C_  A )
10 simprr 756 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  ph )
11 isf32lem.a . . . . . . 7  |-  ( ph  ->  F : om --> ~P G
)
12 isf32lem.b . . . . . . 7  |-  ( ph  ->  A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x ) )
13 isf32lem.c . . . . . . 7  |-  ( ph  ->  -.  |^| ran  F  e. 
ran  F )
1411, 12, 13isf32lem1 8745 . . . . . 6  |-  ( ( ( A  e.  om  /\ 
suc  B  e.  om )  /\  ( suc  B  C_  A  /\  ph )
)  ->  ( F `  A )  C_  ( F `  suc  B ) )
152, 4, 9, 10, 14syl22anc 1229 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( F `  A
)  C_  ( F `  suc  B ) )
1615sseld 3508 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( a  e.  ( F `  A )  ->  a  e.  ( F `  suc  B
) ) )
17 elndif 3633 . . . 4  |-  ( a  e.  ( F `  suc  B )  ->  -.  a  e.  ( ( F `  B )  \  ( F `  suc  B ) ) )
181, 16, 17syl56 34 . . 3  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( a  e.  ( ( F `  A
)  \  ( F `  suc  A ) )  ->  -.  a  e.  ( ( F `  B )  \  ( F `  suc  B ) ) ) )
1918ralrimiv 2879 . 2  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  A. a  e.  (
( F `  A
)  \  ( F `  suc  A ) )  -.  a  e.  ( ( F `  B
)  \  ( F `  suc  B ) ) )
20 disj 3872 . 2  |-  ( ( ( ( F `  A )  \  ( F `  suc  A ) )  i^i  ( ( F `  B ) 
\  ( F `  suc  B ) ) )  =  (/)  <->  A. a  e.  ( ( F `  A
)  \  ( F `  suc  A ) )  -.  a  e.  ( ( F `  B
)  \  ( F `  suc  B ) ) )
2119, 20sylibr 212 1  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( ( ( F `
 A )  \ 
( F `  suc  A ) )  i^i  (
( F `  B
)  \  ( F `  suc  B ) ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817    \ cdif 3478    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   |^|cint 4288   Ord word 4883   suc csuc 4886   ran crn 5006   -->wf 5590   ` cfv 5594   omcom 6695
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-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-iota 5557  df-fv 5602  df-om 6696
This theorem is referenced by:  isf32lem4  8748
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