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Theorem isf32lem3 7981
Description: Lemma for isfin3-2 7993. 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 3298 . . . 4  |-  ( a  e.  ( ( F `
 A )  \ 
( F `  suc  A ) )  ->  a  e.  ( F `  A
) )
2 simpll 730 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  A  e.  om )
3 peano2 4676 . . . . . . 7  |-  ( B  e.  om  ->  suc  B  e.  om )
43ad2antlr 707 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  suc  B  e.  om )
5 nnord 4664 . . . . . . . 8  |-  ( A  e.  om  ->  Ord  A )
65ad2antrr 706 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  Ord  A )
7 simprl 732 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  B  e.  A )
8 ordsucss 4609 . . . . . . 7  |-  ( Ord 
A  ->  ( B  e.  A  ->  suc  B  C_  A ) )
96, 7, 8sylc 56 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  suc  B  C_  A )
10 simprr 733 . . . . . 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 7979 . . . . . 6  |-  ( ( ( A  e.  om  /\ 
suc  B  e.  om )  /\  ( suc  B  C_  A  /\  ph )
)  ->  ( F `  A )  C_  ( F `  suc  B ) )
152, 4, 9, 10, 14syl22anc 1183 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( F `  A
)  C_  ( F `  suc  B ) )
1615sseld 3179 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( a  e.  ( F `  A )  ->  a  e.  ( F `  suc  B
) ) )
17 elndif 3300 . . . 4  |-  ( a  e.  ( F `  suc  B )  ->  -.  a  e.  ( ( F `  B )  \  ( F `  suc  B ) ) )
181, 16, 17syl56 30 . . 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 2625 . 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 3495 . 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 203 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 358    = wceq 1623    e. wcel 1684   A.wral 2543    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   |^|cint 3862   Ord word 4391   suc csuc 4394   omcom 4656   ran crn 4690   -->wf 5251   ` cfv 5255
This theorem is referenced by:  isf32lem4  7982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-iota 5219  df-fv 5263
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