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Theorem sucdom2 7503
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 7333 . . 3  |-  ( A 
~<  B  ->  A  ~<_  B )
2 brdomi 7317 . . 3  |-  ( A  ~<_  B  ->  E. f 
f : A -1-1-> B
)
31, 2syl 16 . 2  |-  ( A 
~<  B  ->  E. f 
f : A -1-1-> B
)
4 relsdom 7313 . . . . . . 7  |-  Rel  ~<
54brrelexi 4875 . . . . . 6  |-  ( A 
~<  B  ->  A  e. 
_V )
65adantr 462 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  A  e.  _V )
7 vex 2973 . . . . . . 7  |-  f  e. 
_V
87rnex 6511 . . . . . 6  |-  ran  f  e.  _V
98a1i 11 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ran  f  e.  _V )
10 f1f1orn 5649 . . . . . . 7  |-  ( f : A -1-1-> B  -> 
f : A -1-1-onto-> ran  f
)
1110adantl 463 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  f : A -1-1-onto-> ran  f )
12 f1of1 5637 . . . . . 6  |-  ( f : A -1-1-onto-> ran  f  ->  f : A -1-1-> ran  f )
1311, 12syl 16 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  f : A -1-1-> ran  f )
14 f1dom2g 7323 . . . . 5  |-  ( ( A  e.  _V  /\  ran  f  e.  _V  /\  f : A -1-1-> ran  f )  ->  A  ~<_  ran  f )
156, 9, 13, 14syl3anc 1213 . . . 4  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  A  ~<_  ran  f
)
16 sdomnen 7334 . . . . . . . 8  |-  ( A 
~<  B  ->  -.  A  ~~  B )
1716adantr 462 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  -.  A  ~~  B )
18 ssdif0 3734 . . . . . . . 8  |-  ( B 
C_  ran  f  <->  ( B  \  ran  f )  =  (/) )
19 simplr 749 . . . . . . . . . . 11  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  f : A -1-1-> B )
20 f1f 5603 . . . . . . . . . . . . . . 15  |-  ( f : A -1-1-> B  -> 
f : A --> B )
21 df-f 5419 . . . . . . . . . . . . . . 15  |-  ( f : A --> B  <->  ( f  Fn  A  /\  ran  f  C_  B ) )
2220, 21sylib 196 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-> B  -> 
( f  Fn  A  /\  ran  f  C_  B
) )
2322simprd 460 . . . . . . . . . . . . 13  |-  ( f : A -1-1-> B  ->  ran  f  C_  B )
2419, 23syl 16 . . . . . . . . . . . 12  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  ran  f  C_  B )
25 simpr 458 . . . . . . . . . . . 12  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  B  C_ 
ran  f )
2624, 25eqssd 3370 . . . . . . . . . . 11  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  ran  f  =  B )
27 dff1o5 5647 . . . . . . . . . . 11  |-  ( f : A -1-1-onto-> B  <->  ( f : A -1-1-> B  /\  ran  f  =  B ) )
2819, 26, 27sylanbrc 659 . . . . . . . . . 10  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  f : A -1-1-onto-> B )
29 f1oen3g 7321 . . . . . . . . . 10  |-  ( ( f  e.  _V  /\  f : A -1-1-onto-> B )  ->  A  ~~  B )
307, 28, 29sylancr 658 . . . . . . . . 9  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  A  ~~  B )
3130ex 434 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( B  C_  ran  f  ->  A  ~~  B ) )
3218, 31syl5bir 218 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( ( B 
\  ran  f )  =  (/)  ->  A  ~~  B ) )
3317, 32mtod 177 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  -.  ( B  \  ran  f )  =  (/) )
34 neq0 3644 . . . . . 6  |-  ( -.  ( B  \  ran  f )  =  (/)  <->  E. w  w  e.  ( B  \  ran  f ) )
3533, 34sylib 196 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  E. w  w  e.  ( B  \  ran  f ) )
36 snssi 4014 . . . . . . 7  |-  ( w  e.  ( B  \  ran  f )  ->  { w }  C_  ( B  \  ran  f ) )
37 vex 2973 . . . . . . . . 9  |-  w  e. 
_V
38 en2sn 7385 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  w  e.  _V )  ->  { A }  ~~  { w } )
396, 37, 38sylancl 657 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  { A }  ~~  { w } )
404brrelex2i 4876 . . . . . . . . . 10  |-  ( A 
~<  B  ->  B  e. 
_V )
4140adantr 462 . . . . . . . . 9  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  B  e.  _V )
42 difexg 4437 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( B  \  ran  f )  e.  _V )
43 ssdomg 7351 . . . . . . . . 9  |-  ( ( B  \  ran  f
)  e.  _V  ->  ( { w }  C_  ( B  \  ran  f
)  ->  { w }  ~<_  ( B  \  ran  f ) ) )
4441, 42, 433syl 20 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( { w }  C_  ( B  \  ran  f )  ->  { w }  ~<_  ( B  \  ran  f ) ) )
45 endomtr 7363 . . . . . . . 8  |-  ( ( { A }  ~~  { w }  /\  {
w }  ~<_  ( B 
\  ran  f )
)  ->  { A }  ~<_  ( B  \  ran  f ) )
4639, 44, 45syl6an 542 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( { w }  C_  ( B  \  ran  f )  ->  { A }  ~<_  ( B  \  ran  f ) ) )
4736, 46syl5 32 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( w  e.  ( B  \  ran  f )  ->  { A }  ~<_  ( B  \  ran  f ) ) )
4847exlimdv 1695 . . . . 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 3748 . . . . 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 7395 . . . 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 1212 . . 3  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( A  u.  { A } )  ~<_  ( ran  f  u.  ( B  \  ran  f ) ) )
54 df-suc 4721 . . . 4  |-  suc  A  =  ( A  u.  { A } )
5554a1i 11 . . 3  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  suc  A  =  ( A  u.  { A } ) )
56 undif2 3752 . . . 4  |-  ( ran  f  u.  ( B 
\  ran  f )
)  =  ( ran  f  u.  B )
5723adantl 463 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ran  f  C_  B )
58 ssequn1 3523 . . . . 5  |-  ( ran  f  C_  B  <->  ( ran  f  u.  B )  =  B )
5957, 58sylib 196 . . . 4  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( ran  f  u.  B )  =  B )
6056, 59syl5req 2486 . . 3  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  B  =  ( ran  f  u.  ( B  \  ran  f ) ) )
6153, 55, 603brtr4d 4319 . 2  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  suc  A  ~<_  B )
623, 61exlimddv 1697 1  |-  ( A 
~<  B  ->  suc  A  ~<_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1364   E.wex 1591    e. wcel 1761   _Vcvv 2970    \ cdif 3322    u. cun 3323    i^i cin 3324    C_ wss 3325   (/)c0 3634   {csn 3874   class class class wbr 4289   suc csuc 4717   ran crn 4837    Fn wfn 5410   -->wf 5411   -1-1->wf1 5412   -1-1-onto->wf1o 5414    ~~ cen 7303    ~<_ cdom 7304    ~< csdm 7305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-id 4632  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-1o 6916  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309
This theorem is referenced by:  sucdom  7504  card2inf  7766
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