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Theorem sucdom2 7057
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 6889 . . 3  |-  ( A 
~<  B  ->  A  ~<_  B )
2 brdomi 6873 . . 3  |-  ( A  ~<_  B  ->  E. f 
f : A -1-1-> B
)
31, 2syl 15 . 2  |-  ( A 
~<  B  ->  E. f 
f : A -1-1-> B
)
4 relsdom 6870 . . . . . . . . 9  |-  Rel  ~<
54brrelexi 4729 . . . . . . . 8  |-  ( A 
~<  B  ->  A  e. 
_V )
65adantr 451 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  A  e.  _V )
7 vex 2791 . . . . . . . . 9  |-  f  e. 
_V
87rnex 4942 . . . . . . . 8  |-  ran  f  e.  _V
98a1i 10 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ran  f  e.  _V )
10 f1f1orn 5483 . . . . . . . . 9  |-  ( f : A -1-1-> B  -> 
f : A -1-1-onto-> ran  f
)
1110adantl 452 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  f : A -1-1-onto-> ran  f )
12 f1of1 5471 . . . . . . . 8  |-  ( f : A -1-1-onto-> ran  f  ->  f : A -1-1-> ran  f )
1311, 12syl 15 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  f : A -1-1-> ran  f )
14 f1dom2g 6879 . . . . . . 7  |-  ( ( A  e.  _V  /\  ran  f  e.  _V  /\  f : A -1-1-> ran  f )  ->  A  ~<_  ran  f )
156, 9, 13, 14syl3anc 1182 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  A  ~<_  ran  f
)
16 sdomnen 6890 . . . . . . . . . 10  |-  ( A 
~<  B  ->  -.  A  ~~  B )
1716adantr 451 . . . . . . . . 9  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  -.  A  ~~  B )
18 ssdif0 3513 . . . . . . . . . 10  |-  ( B 
C_  ran  f  <->  ( B  \  ran  f )  =  (/) )
19 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  f : A -1-1-> B )
20 f1f 5437 . . . . . . . . . . . . . . . . 17  |-  ( f : A -1-1-> B  -> 
f : A --> B )
21 df-f 5259 . . . . . . . . . . . . . . . . 17  |-  ( f : A --> B  <->  ( f  Fn  A  /\  ran  f  C_  B ) )
2220, 21sylib 188 . . . . . . . . . . . . . . . 16  |-  ( f : A -1-1-> B  -> 
( f  Fn  A  /\  ran  f  C_  B
) )
2322simprd 449 . . . . . . . . . . . . . . 15  |-  ( f : A -1-1-> B  ->  ran  f  C_  B )
2419, 23syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  ran  f  C_  B )
25 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  B  C_ 
ran  f )
2624, 25eqssd 3196 . . . . . . . . . . . . 13  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  ran  f  =  B )
27 dff1o5 5481 . . . . . . . . . . . . 13  |-  ( f : A -1-1-onto-> B  <->  ( f : A -1-1-> B  /\  ran  f  =  B ) )
2819, 26, 27sylanbrc 645 . . . . . . . . . . . 12  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  f : A -1-1-onto-> B )
29 f1oen3g 6877 . . . . . . . . . . . 12  |-  ( ( f  e.  _V  /\  f : A -1-1-onto-> B )  ->  A  ~~  B )
307, 28, 29sylancr 644 . . . . . . . . . . 11  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  A  ~~  B )
3130ex 423 . . . . . . . . . 10  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( B  C_  ran  f  ->  A  ~~  B ) )
3218, 31syl5bir 209 . . . . . . . . 9  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( ( B 
\  ran  f )  =  (/)  ->  A  ~~  B ) )
3317, 32mtod 168 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  -.  ( B  \  ran  f )  =  (/) )
34 neq0 3465 . . . . . . . 8  |-  ( -.  ( B  \  ran  f )  =  (/)  <->  E. w  w  e.  ( B  \  ran  f ) )
3533, 34sylib 188 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  E. w  w  e.  ( B  \  ran  f ) )
36 snssi 3759 . . . . . . . . 9  |-  ( w  e.  ( B  \  ran  f )  ->  { w }  C_  ( B  \  ran  f ) )
37 vex 2791 . . . . . . . . . . 11  |-  w  e. 
_V
38 en2sn 6940 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  w  e.  _V )  ->  { A }  ~~  { w } )
396, 37, 38sylancl 643 . . . . . . . . . 10  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  { A }  ~~  { w } )
404brrelex2i 4730 . . . . . . . . . . . 12  |-  ( A 
~<  B  ->  B  e. 
_V )
4140adantr 451 . . . . . . . . . . 11  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  B  e.  _V )
42 difexg 4162 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( B  \  ran  f )  e.  _V )
43 ssdomg 6907 . . . . . . . . . . 11  |-  ( ( B  \  ran  f
)  e.  _V  ->  ( { w }  C_  ( B  \  ran  f
)  ->  { w }  ~<_  ( B  \  ran  f ) ) )
4441, 42, 433syl 18 . . . . . . . . . 10  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( { w }  C_  ( B  \  ran  f )  ->  { w }  ~<_  ( B  \  ran  f ) ) )
45 endomtr 6919 . . . . . . . . . 10  |-  ( ( { A }  ~~  { w }  /\  {
w }  ~<_  ( B 
\  ran  f )
)  ->  { A }  ~<_  ( B  \  ran  f ) )
4639, 44, 45ee12an 1353 . . . . . . . . 9  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( { w }  C_  ( B  \  ran  f )  ->  { A }  ~<_  ( B  \  ran  f ) ) )
4736, 46syl5 28 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( w  e.  ( B  \  ran  f )  ->  { A }  ~<_  ( B  \  ran  f ) ) )
4847exlimdv 1664 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( E. w  w  e.  ( B  \  ran  f )  ->  { A }  ~<_  ( B 
\  ran  f )
) )
4935, 48mpd 14 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  { A }  ~<_  ( B  \  ran  f
) )
50 disjdif 3526 . . . . . . 7  |-  ( ran  f  i^i  ( B 
\  ran  f )
)  =  (/)
5150a1i 10 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( ran  f  i^i  ( B  \  ran  f ) )  =  (/) )
52 undom 6950 . . . . . 6  |-  ( ( ( 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 1181 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( A  u.  { A } )  ~<_  ( ran  f  u.  ( B  \  ran  f ) ) )
54 df-suc 4398 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
5554a1i 10 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  suc  A  =  ( A  u.  { A } ) )
56 undif2 3530 . . . . . 6  |-  ( ran  f  u.  ( B 
\  ran  f )
)  =  ( ran  f  u.  B )
5723adantl 452 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ran  f  C_  B )
58 ssequn1 3345 . . . . . . 7  |-  ( ran  f  C_  B  <->  ( ran  f  u.  B )  =  B )
5957, 58sylib 188 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( ran  f  u.  B )  =  B )
6056, 59syl5req 2328 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  B  =  ( ran  f  u.  ( B  \  ran  f ) ) )
6153, 55, 603brtr4d 4053 . . . 4  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  suc  A  ~<_  B )
6261ex 423 . . 3  |-  ( A 
~<  B  ->  ( f : A -1-1-> B  ->  suc  A  ~<_  B ) )
6362exlimdv 1664 . 2  |-  ( A 
~<  B  ->  ( E. f  f : A -1-1-> B  ->  suc  A  ~<_  B ) )
643, 63mpd 14 1  |-  ( A 
~<  B  ->  suc  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023   suc csuc 4394   ran crn 4690    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  sucdom  7058  card2inf  7269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866
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