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Theorem isf32lem3 8240
Description: Lemma for isfin3-2 8252. 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 3471 . . . 4  |-  ( a  e.  ( ( F `
 A )  \ 
( F `  suc  A ) )  ->  a  e.  ( F `  A
) )
2 simpll 732 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  A  e.  om )
3 peano2 4868 . . . . . . 7  |-  ( B  e.  om  ->  suc  B  e.  om )
43ad2antlr 709 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  suc  B  e.  om )
5 nnord 4856 . . . . . . . 8  |-  ( A  e.  om  ->  Ord  A )
65ad2antrr 708 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  Ord  A )
7 simprl 734 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  B  e.  A )
8 ordsucss 4801 . . . . . . 7  |-  ( Ord 
A  ->  ( B  e.  A  ->  suc  B  C_  A ) )
96, 7, 8sylc 59 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  suc  B  C_  A )
10 simprr 735 . . . . . 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 8238 . . . . . 6  |-  ( ( ( A  e.  om  /\ 
suc  B  e.  om )  /\  ( suc  B  C_  A  /\  ph )
)  ->  ( F `  A )  C_  ( F `  suc  B ) )
152, 4, 9, 10, 14syl22anc 1186 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( F `  A
)  C_  ( F `  suc  B ) )
1615sseld 3349 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( a  e.  ( F `  A )  ->  a  e.  ( F `  suc  B
) ) )
17 elndif 3473 . . . 4  |-  ( a  e.  ( F `  suc  B )  ->  -.  a  e.  ( ( F `  B )  \  ( F `  suc  B ) ) )
181, 16, 17syl56 33 . . 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 2790 . 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 3670 . 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 205 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   |^|cint 4052   Ord word 4583   suc csuc 4586   omcom 4848   ran crn 4882   -->wf 5453   ` cfv 5457
This theorem is referenced by:  isf32lem4  8241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-iota 5421  df-fv 5465
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