MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isf32lem4 Unicode version

Theorem isf32lem4 7998
Description: Lemma for isfin3-2 8009. Being a chain, difference sets are disjoint. (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
isf32lem4  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( ( ( 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 isf32lem4
StepHypRef Expression
1 simplrr 737 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  B  e.  om )
2 simplrl 736 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  A  e.  om )
3 simpr 447 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  A  e.  B )
4 simplll 734 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  ph )
5 incom 3374 . . . 4  |-  ( ( ( F `  A
)  \  ( F `  suc  A ) )  i^i  ( ( F `
 B )  \ 
( F `  suc  B ) ) )  =  ( ( ( F `
 B )  \ 
( F `  suc  B ) )  i^i  (
( F `  A
)  \  ( F `  suc  A ) ) )
6 isf32lem.a . . . . 5  |-  ( ph  ->  F : om --> ~P G
)
7 isf32lem.b . . . . 5  |-  ( ph  ->  A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x ) )
8 isf32lem.c . . . . 5  |-  ( ph  ->  -.  |^| ran  F  e. 
ran  F )
96, 7, 8isf32lem3 7997 . . . 4  |-  ( ( ( B  e.  om  /\  A  e.  om )  /\  ( A  e.  B  /\  ph ) )  -> 
( ( ( F `
 B )  \ 
( F `  suc  B ) )  i^i  (
( F `  A
)  \  ( F `  suc  A ) ) )  =  (/) )
105, 9syl5eq 2340 . . 3  |-  ( ( ( B  e.  om  /\  A  e.  om )  /\  ( A  e.  B  /\  ph ) )  -> 
( ( ( F `
 A )  \ 
( F `  suc  A ) )  i^i  (
( F `  B
)  \  ( F `  suc  B ) ) )  =  (/) )
111, 2, 3, 4, 10syl22anc 1183 . 2  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  A  e.  B )  ->  (
( ( F `  A )  \  ( F `  suc  A ) )  i^i  ( ( F `  B ) 
\  ( F `  suc  B ) ) )  =  (/) )
12 simplrl 736 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  A  e.  om )
13 simplrr 737 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  B  e.  om )
14 simpr 447 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  B  e.  A )
15 simplll 734 . . 3  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  ph )
166, 7, 8isf32lem3 7997 . . 3  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  e.  A  /\  ph ) )  -> 
( ( ( F `
 A )  \ 
( F `  suc  A ) )  i^i  (
( F `  B
)  \  ( F `  suc  B ) ) )  =  (/) )
1712, 13, 14, 15, 16syl22anc 1183 . 2  |-  ( ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om )
)  /\  B  e.  A )  ->  (
( ( F `  A )  \  ( F `  suc  A ) )  i^i  ( ( F `  B ) 
\  ( F `  suc  B ) ) )  =  (/) )
18 simplr 731 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  ->  A  =/=  B )
19 nnord 4680 . . . . . 6  |-  ( A  e.  om  ->  Ord  A )
20 nnord 4680 . . . . . 6  |-  ( B  e.  om  ->  Ord  B )
21 ordtri3 4444 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A ) ) )
2219, 20, 21syl2an 463 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A
) ) )
2322adantl 452 . . . 4  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( A  =  B  <->  -.  ( A  e.  B  \/  B  e.  A
) ) )
2423necon2abid 2516 . . 3  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( ( A  e.  B  \/  B  e.  A )  <->  A  =/=  B ) )
2518, 24mpbird 223 . 2  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( A  e.  B  \/  B  e.  A
) )
2611, 17, 25mpjaodan 761 1  |-  ( ( ( ph  /\  A  =/=  B )  /\  ( A  e.  om  /\  B  e.  om ) )  -> 
( ( ( F `
 A )  \ 
( F `  suc  A ) )  i^i  (
( F `  B
)  \  ( F `  suc  B ) ) )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   |^|cint 3878   Ord word 4407   suc csuc 4410   omcom 4672   ran crn 4706   -->wf 5267   ` cfv 5271
This theorem is referenced by:  isf32lem7  8001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-iota 5235  df-fv 5279
  Copyright terms: Public domain W3C validator