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Theorem isf32lem3 8736
Description: Lemma for isfin3-2 8748. 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 3530 . . . 4  |-  ( a  e.  ( ( F `
 A )  \ 
( F `  suc  A ) )  ->  a  e.  ( F `  A
) )
2 simpll 758 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  A  e.  om )
3 peano2 6671 . . . . . . 7  |-  ( B  e.  om  ->  suc  B  e.  om )
43ad2antlr 731 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  suc  B  e.  om )
5 nnord 6658 . . . . . . . 8  |-  ( A  e.  om  ->  Ord  A )
65ad2antrr 730 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  Ord  A )
7 simprl 762 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  B  e.  A )
8 ordsucss 6603 . . . . . . 7  |-  ( Ord 
A  ->  ( B  e.  A  ->  suc  B  C_  A ) )
96, 7, 8sylc 62 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  suc  B  C_  A )
10 simprr 764 . . . . . 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 8734 . . . . . 6  |-  ( ( ( A  e.  om  /\ 
suc  B  e.  om )  /\  ( suc  B  C_  A  /\  ph )
)  ->  ( F `  A )  C_  ( F `  suc  B ) )
152, 4, 9, 10, 14syl22anc 1265 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( F `  A
)  C_  ( F `  suc  B ) )
1615sseld 3406 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( a  e.  ( F `  A )  ->  a  e.  ( F `  suc  B
) ) )
17 elndif 3532 . . . 4  |-  ( a  e.  ( F `  suc  B )  ->  -.  a  e.  ( ( F `  B )  \  ( F `  suc  B ) ) )
181, 16, 17syl56 35 . . 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 2777 . 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 3778 . 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 215 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 370    = wceq 1437    e. wcel 1872   A.wral 2714    \ cdif 3376    i^i cin 3378    C_ wss 3379   (/)c0 3704   ~Pcpw 3924   |^|cint 4198   ran crn 4797   Ord word 5384   suc csuc 5387   -->wf 5540   ` cfv 5544   omcom 6650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541
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 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-tr 4462  df-eprel 4707  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fv 5552  df-om 6651
This theorem is referenced by:  isf32lem4  8737
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