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Theorem sucdom2 7768
Description: Strict dominance of a set over another set implies dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
sucdom2  |-  ( A 
~<  B  ->  suc  A  ~<_  B )

Proof of Theorem sucdom2
Dummy variables  w  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sdomdom 7597 . . 3  |-  ( A 
~<  B  ->  A  ~<_  B )
2 brdomi 7580 . . 3  |-  ( A  ~<_  B  ->  E. f 
f : A -1-1-> B
)
31, 2syl 17 . 2  |-  ( A 
~<  B  ->  E. f 
f : A -1-1-> B
)
4 relsdom 7576 . . . . . . 7  |-  Rel  ~<
54brrelexi 4875 . . . . . 6  |-  ( A 
~<  B  ->  A  e. 
_V )
65adantr 467 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  A  e.  _V )
7 vex 3048 . . . . . . 7  |-  f  e. 
_V
87rnex 6727 . . . . . 6  |-  ran  f  e.  _V
98a1i 11 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ran  f  e.  _V )
10 f1f1orn 5825 . . . . . . 7  |-  ( f : A -1-1-> B  -> 
f : A -1-1-onto-> ran  f
)
1110adantl 468 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  f : A -1-1-onto-> ran  f )
12 f1of1 5813 . . . . . 6  |-  ( f : A -1-1-onto-> ran  f  ->  f : A -1-1-> ran  f )
1311, 12syl 17 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  f : A -1-1-> ran  f )
14 f1dom2g 7587 . . . . 5  |-  ( ( A  e.  _V  /\  ran  f  e.  _V  /\  f : A -1-1-> ran  f )  ->  A  ~<_  ran  f )
156, 9, 13, 14syl3anc 1268 . . . 4  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  A  ~<_  ran  f
)
16 sdomnen 7598 . . . . . . . 8  |-  ( A 
~<  B  ->  -.  A  ~~  B )
1716adantr 467 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  -.  A  ~~  B )
18 ssdif0 3823 . . . . . . . 8  |-  ( B 
C_  ran  f  <->  ( B  \  ran  f )  =  (/) )
19 simplr 762 . . . . . . . . . . 11  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  f : A -1-1-> B )
20 f1f 5779 . . . . . . . . . . . . . . 15  |-  ( f : A -1-1-> B  -> 
f : A --> B )
21 df-f 5586 . . . . . . . . . . . . . . 15  |-  ( f : A --> B  <->  ( f  Fn  A  /\  ran  f  C_  B ) )
2220, 21sylib 200 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-> B  -> 
( f  Fn  A  /\  ran  f  C_  B
) )
2322simprd 465 . . . . . . . . . . . . 13  |-  ( f : A -1-1-> B  ->  ran  f  C_  B )
2419, 23syl 17 . . . . . . . . . . . 12  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  ran  f  C_  B )
25 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  B  C_ 
ran  f )
2624, 25eqssd 3449 . . . . . . . . . . 11  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  ran  f  =  B )
27 dff1o5 5823 . . . . . . . . . . 11  |-  ( f : A -1-1-onto-> B  <->  ( f : A -1-1-> B  /\  ran  f  =  B ) )
2819, 26, 27sylanbrc 670 . . . . . . . . . 10  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  f : A -1-1-onto-> B )
29 f1oen3g 7585 . . . . . . . . . 10  |-  ( ( f  e.  _V  /\  f : A -1-1-onto-> B )  ->  A  ~~  B )
307, 28, 29sylancr 669 . . . . . . . . 9  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  A  ~~  B )
3130ex 436 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( B  C_  ran  f  ->  A  ~~  B ) )
3218, 31syl5bir 222 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( ( B 
\  ran  f )  =  (/)  ->  A  ~~  B ) )
3317, 32mtod 181 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  -.  ( B  \  ran  f )  =  (/) )
34 neq0 3742 . . . . . 6  |-  ( -.  ( B  \  ran  f )  =  (/)  <->  E. w  w  e.  ( B  \  ran  f ) )
3533, 34sylib 200 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  E. w  w  e.  ( B  \  ran  f ) )
36 snssi 4116 . . . . . . 7  |-  ( w  e.  ( B  \  ran  f )  ->  { w }  C_  ( B  \  ran  f ) )
37 vex 3048 . . . . . . . . 9  |-  w  e. 
_V
38 en2sn 7649 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  w  e.  _V )  ->  { A }  ~~  { w } )
396, 37, 38sylancl 668 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  { A }  ~~  { w } )
404brrelex2i 4876 . . . . . . . . . 10  |-  ( A 
~<  B  ->  B  e. 
_V )
4140adantr 467 . . . . . . . . 9  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  B  e.  _V )
42 difexg 4551 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( B  \  ran  f )  e.  _V )
43 ssdomg 7615 . . . . . . . . 9  |-  ( ( B  \  ran  f
)  e.  _V  ->  ( { w }  C_  ( B  \  ran  f
)  ->  { w }  ~<_  ( B  \  ran  f ) ) )
4441, 42, 433syl 18 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( { w }  C_  ( B  \  ran  f )  ->  { w }  ~<_  ( B  \  ran  f ) ) )
45 endomtr 7627 . . . . . . . 8  |-  ( ( { A }  ~~  { w }  /\  {
w }  ~<_  ( B 
\  ran  f )
)  ->  { A }  ~<_  ( B  \  ran  f ) )
4639, 44, 45syl6an 548 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( { w }  C_  ( B  \  ran  f )  ->  { A }  ~<_  ( B  \  ran  f ) ) )
4736, 46syl5 33 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( w  e.  ( B  \  ran  f )  ->  { A }  ~<_  ( B  \  ran  f ) ) )
4847exlimdv 1779 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( E. w  w  e.  ( B  \  ran  f )  ->  { A }  ~<_  ( B 
\  ran  f )
) )
4935, 48mpd 15 . . . 4  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  { A }  ~<_  ( B  \  ran  f
) )
50 disjdif 3839 . . . . 5  |-  ( ran  f  i^i  ( B 
\  ran  f )
)  =  (/)
5150a1i 11 . . . 4  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( ran  f  i^i  ( B  \  ran  f ) )  =  (/) )
52 undom 7660 . . . 4  |-  ( ( ( A  ~<_  ran  f  /\  { A }  ~<_  ( B 
\  ran  f )
)  /\  ( ran  f  i^i  ( B  \  ran  f ) )  =  (/) )  ->  ( A  u.  { A }
)  ~<_  ( ran  f  u.  ( B  \  ran  f ) ) )
5315, 49, 51, 52syl21anc 1267 . . 3  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( A  u.  { A } )  ~<_  ( ran  f  u.  ( B  \  ran  f ) ) )
54 df-suc 5429 . . . 4  |-  suc  A  =  ( A  u.  { A } )
5554a1i 11 . . 3  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  suc  A  =  ( A  u.  { A } ) )
56 undif2 3843 . . . 4  |-  ( ran  f  u.  ( B 
\  ran  f )
)  =  ( ran  f  u.  B )
5723adantl 468 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ran  f  C_  B )
58 ssequn1 3604 . . . . 5  |-  ( ran  f  C_  B  <->  ( ran  f  u.  B )  =  B )
5957, 58sylib 200 . . . 4  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( ran  f  u.  B )  =  B )
6056, 59syl5req 2498 . . 3  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  B  =  ( ran  f  u.  ( B  \  ran  f ) ) )
6153, 55, 603brtr4d 4433 . 2  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  suc  A  ~<_  B )
623, 61exlimddv 1781 1  |-  ( A 
~<  B  ->  suc  A  ~<_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887   _Vcvv 3045    \ cdif 3401    u. cun 3402    i^i cin 3403    C_ wss 3404   (/)c0 3731   {csn 3968   class class class wbr 4402   ran crn 4835   suc csuc 5425    Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   -1-1-onto->wf1o 5581    ~~ cen 7566    ~<_ cdom 7567    ~< csdm 7568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-suc 5429  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572
This theorem is referenced by:  sucdom  7769  card2inf  8070
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