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Theorem isf32lem3 8191
Description: Lemma for isfin3-2 8203. 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 3429 . . . 4  |-  ( a  e.  ( ( F `
 A )  \ 
( F `  suc  A ) )  ->  a  e.  ( F `  A
) )
2 simpll 731 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  A  e.  om )
3 peano2 4824 . . . . . . 7  |-  ( B  e.  om  ->  suc  B  e.  om )
43ad2antlr 708 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  suc  B  e.  om )
5 nnord 4812 . . . . . . . 8  |-  ( A  e.  om  ->  Ord  A )
65ad2antrr 707 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  Ord  A )
7 simprl 733 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  B  e.  A )
8 ordsucss 4757 . . . . . . 7  |-  ( Ord 
A  ->  ( B  e.  A  ->  suc  B  C_  A ) )
96, 7, 8sylc 58 . . . . . 6  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  ->  suc  B  C_  A )
10 simprr 734 . . . . . 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 8189 . . . . . 6  |-  ( ( ( A  e.  om  /\ 
suc  B  e.  om )  /\  ( suc  B  C_  A  /\  ph )
)  ->  ( F `  A )  C_  ( F `  suc  B ) )
152, 4, 9, 10, 14syl22anc 1185 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( F `  A
)  C_  ( F `  suc  B ) )
1615sseld 3307 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( a  e.  ( F `  A )  ->  a  e.  ( F `  suc  B
) ) )
17 elndif 3431 . . . 4  |-  ( a  e.  ( F `  suc  B )  ->  -.  a  e.  ( ( F `  B )  \  ( F `  suc  B ) ) )
181, 16, 17syl56 32 . . 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 2748 . 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 3628 . 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 204 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 359    = wceq 1649    e. wcel 1721   A.wral 2666    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   |^|cint 4010   Ord word 4540   suc csuc 4543   omcom 4804   ran crn 4838   -->wf 5409   ` cfv 5413
This theorem is referenced by:  isf32lem4  8192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-iota 5377  df-fv 5421
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