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Theorem sucdom2 7303
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 7135 . . 3  |-  ( A 
~<  B  ->  A  ~<_  B )
2 brdomi 7119 . . 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 7116 . . . . . . 7  |-  Rel  ~<
54brrelexi 4918 . . . . . 6  |-  ( A 
~<  B  ->  A  e. 
_V )
65adantr 452 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  A  e.  _V )
7 vex 2959 . . . . . . 7  |-  f  e. 
_V
87rnex 5133 . . . . . 6  |-  ran  f  e.  _V
98a1i 11 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ran  f  e.  _V )
10 f1f1orn 5685 . . . . . . 7  |-  ( f : A -1-1-> B  -> 
f : A -1-1-onto-> ran  f
)
1110adantl 453 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  f : A -1-1-onto-> ran  f )
12 f1of1 5673 . . . . . 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 7125 . . . . 5  |-  ( ( A  e.  _V  /\  ran  f  e.  _V  /\  f : A -1-1-> ran  f )  ->  A  ~<_  ran  f )
156, 9, 13, 14syl3anc 1184 . . . 4  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  A  ~<_  ran  f
)
16 sdomnen 7136 . . . . . . . 8  |-  ( A 
~<  B  ->  -.  A  ~~  B )
1716adantr 452 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  -.  A  ~~  B )
18 ssdif0 3686 . . . . . . . 8  |-  ( B 
C_  ran  f  <->  ( B  \  ran  f )  =  (/) )
19 simplr 732 . . . . . . . . . . 11  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  f : A -1-1-> B )
20 f1f 5639 . . . . . . . . . . . . . . 15  |-  ( f : A -1-1-> B  -> 
f : A --> B )
21 df-f 5458 . . . . . . . . . . . . . . 15  |-  ( f : A --> B  <->  ( f  Fn  A  /\  ran  f  C_  B ) )
2220, 21sylib 189 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-> B  -> 
( f  Fn  A  /\  ran  f  C_  B
) )
2322simprd 450 . . . . . . . . . . . . 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 448 . . . . . . . . . . . 12  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  B  C_ 
ran  f )
2624, 25eqssd 3365 . . . . . . . . . . 11  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  ran  f  =  B )
27 dff1o5 5683 . . . . . . . . . . 11  |-  ( f : A -1-1-onto-> B  <->  ( f : A -1-1-> B  /\  ran  f  =  B ) )
2819, 26, 27sylanbrc 646 . . . . . . . . . 10  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  f : A -1-1-onto-> B )
29 f1oen3g 7123 . . . . . . . . . 10  |-  ( ( f  e.  _V  /\  f : A -1-1-onto-> B )  ->  A  ~~  B )
307, 28, 29sylancr 645 . . . . . . . . 9  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  A  ~~  B )
3130ex 424 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( B  C_  ran  f  ->  A  ~~  B ) )
3218, 31syl5bir 210 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( ( B 
\  ran  f )  =  (/)  ->  A  ~~  B ) )
3317, 32mtod 170 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  -.  ( B  \  ran  f )  =  (/) )
34 neq0 3638 . . . . . 6  |-  ( -.  ( B  \  ran  f )  =  (/)  <->  E. w  w  e.  ( B  \  ran  f ) )
3533, 34sylib 189 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  E. w  w  e.  ( B  \  ran  f ) )
36 snssi 3942 . . . . . . 7  |-  ( w  e.  ( B  \  ran  f )  ->  { w }  C_  ( B  \  ran  f ) )
37 vex 2959 . . . . . . . . 9  |-  w  e. 
_V
38 en2sn 7186 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  w  e.  _V )  ->  { A }  ~~  { w } )
396, 37, 38sylancl 644 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  { A }  ~~  { w } )
404brrelex2i 4919 . . . . . . . . . 10  |-  ( A 
~<  B  ->  B  e. 
_V )
4140adantr 452 . . . . . . . . 9  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  B  e.  _V )
42 difexg 4351 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( B  \  ran  f )  e.  _V )
43 ssdomg 7153 . . . . . . . . 9  |-  ( ( B  \  ran  f
)  e.  _V  ->  ( { w }  C_  ( B  \  ran  f
)  ->  { w }  ~<_  ( B  \  ran  f ) ) )
4441, 42, 433syl 19 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( { w }  C_  ( B  \  ran  f )  ->  { w }  ~<_  ( B  \  ran  f ) ) )
45 endomtr 7165 . . . . . . . 8  |-  ( ( { A }  ~~  { w }  /\  {
w }  ~<_  ( B 
\  ran  f )
)  ->  { A }  ~<_  ( B  \  ran  f ) )
4639, 44, 45ee12an 1372 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( { w }  C_  ( B  \  ran  f )  ->  { A }  ~<_  ( B  \  ran  f ) ) )
4736, 46syl5 30 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( w  e.  ( B  \  ran  f )  ->  { A }  ~<_  ( B  \  ran  f ) ) )
4847exlimdv 1646 . . . . 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 3700 . . . . 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 7196 . . . 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 1183 . . 3  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( A  u.  { A } )  ~<_  ( ran  f  u.  ( B  \  ran  f ) ) )
54 df-suc 4587 . . . 4  |-  suc  A  =  ( A  u.  { A } )
5554a1i 11 . . 3  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  suc  A  =  ( A  u.  { A } ) )
56 undif2 3704 . . . 4  |-  ( ran  f  u.  ( B 
\  ran  f )
)  =  ( ran  f  u.  B )
5723adantl 453 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ran  f  C_  B )
58 ssequn1 3517 . . . . 5  |-  ( ran  f  C_  B  <->  ( ran  f  u.  B )  =  B )
5957, 58sylib 189 . . . 4  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( ran  f  u.  B )  =  B )
6056, 59syl5req 2481 . . 3  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  B  =  ( ran  f  u.  ( B  \  ran  f ) ) )
6153, 55, 603brtr4d 4242 . 2  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  suc  A  ~<_  B )
623, 61exlimddv 1648 1  |-  ( A 
~<  B  ->  suc  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   {csn 3814   class class class wbr 4212   suc csuc 4583   ran crn 4879    Fn wfn 5449   -->wf 5450   -1-1->wf1 5451   -1-1-onto->wf1o 5453    ~~ cen 7106    ~<_ cdom 7107    ~< csdm 7108
This theorem is referenced by:  sucdom  7304  card2inf  7523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112
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