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Theorem sucdom2 7526
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 7356 . . 3  |-  ( A 
~<  B  ->  A  ~<_  B )
2 brdomi 7340 . . 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 7336 . . . . . . 7  |-  Rel  ~<
54brrelexi 4898 . . . . . 6  |-  ( A 
~<  B  ->  A  e. 
_V )
65adantr 465 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  A  e.  _V )
7 vex 2994 . . . . . . 7  |-  f  e. 
_V
87rnex 6531 . . . . . 6  |-  ran  f  e.  _V
98a1i 11 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ran  f  e.  _V )
10 f1f1orn 5671 . . . . . . 7  |-  ( f : A -1-1-> B  -> 
f : A -1-1-onto-> ran  f
)
1110adantl 466 . . . . . 6  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  f : A -1-1-onto-> ran  f )
12 f1of1 5659 . . . . . 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 7346 . . . . 5  |-  ( ( A  e.  _V  /\  ran  f  e.  _V  /\  f : A -1-1-> ran  f )  ->  A  ~<_  ran  f )
156, 9, 13, 14syl3anc 1218 . . . 4  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  A  ~<_  ran  f
)
16 sdomnen 7357 . . . . . . . 8  |-  ( A 
~<  B  ->  -.  A  ~~  B )
1716adantr 465 . . . . . . 7  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  -.  A  ~~  B )
18 ssdif0 3756 . . . . . . . 8  |-  ( B 
C_  ran  f  <->  ( B  \  ran  f )  =  (/) )
19 simplr 754 . . . . . . . . . . 11  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  f : A -1-1-> B )
20 f1f 5625 . . . . . . . . . . . . . . 15  |-  ( f : A -1-1-> B  -> 
f : A --> B )
21 df-f 5441 . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . 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 461 . . . . . . . . . . . 12  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  B  C_ 
ran  f )
2624, 25eqssd 3392 . . . . . . . . . . 11  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  ran  f  =  B )
27 dff1o5 5669 . . . . . . . . . . 11  |-  ( f : A -1-1-onto-> B  <->  ( f : A -1-1-> B  /\  ran  f  =  B ) )
2819, 26, 27sylanbrc 664 . . . . . . . . . 10  |-  ( ( ( A  ~<  B  /\  f : A -1-1-> B )  /\  B  C_  ran  f )  ->  f : A -1-1-onto-> B )
29 f1oen3g 7344 . . . . . . . . . 10  |-  ( ( f  e.  _V  /\  f : A -1-1-onto-> B )  ->  A  ~~  B )
307, 28, 29sylancr 663 . . . . . . . . 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 3666 . . . . . 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 4036 . . . . . . 7  |-  ( w  e.  ( B  \  ran  f )  ->  { w }  C_  ( B  \  ran  f ) )
37 vex 2994 . . . . . . . . 9  |-  w  e. 
_V
38 en2sn 7408 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  w  e.  _V )  ->  { A }  ~~  { w } )
396, 37, 38sylancl 662 . . . . . . . 8  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  { A }  ~~  { w } )
404brrelex2i 4899 . . . . . . . . . 10  |-  ( A 
~<  B  ->  B  e. 
_V )
4140adantr 465 . . . . . . . . 9  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  B  e.  _V )
42 difexg 4459 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( B  \  ran  f )  e.  _V )
43 ssdomg 7374 . . . . . . . . 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 7386 . . . . . . . 8  |-  ( ( { A }  ~~  { w }  /\  {
w }  ~<_  ( B 
\  ran  f )
)  ->  { A }  ~<_  ( B  \  ran  f ) )
4639, 44, 45syl6an 545 . . . . . . 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 1690 . . . . 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 3770 . . . . 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 7418 . . . 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 1217 . . 3  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ( A  u.  { A } )  ~<_  ( ran  f  u.  ( B  \  ran  f ) ) )
54 df-suc 4744 . . . 4  |-  suc  A  =  ( A  u.  { A } )
5554a1i 11 . . 3  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  suc  A  =  ( A  u.  { A } ) )
56 undif2 3774 . . . 4  |-  ( ran  f  u.  ( B 
\  ran  f )
)  =  ( ran  f  u.  B )
5723adantl 466 . . . . 5  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  ran  f  C_  B )
58 ssequn1 3545 . . . . 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 2488 . . 3  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  B  =  ( ran  f  u.  ( B  \  ran  f ) ) )
6153, 55, 603brtr4d 4341 . 2  |-  ( ( A  ~<  B  /\  f : A -1-1-> B )  ->  suc  A  ~<_  B )
623, 61exlimddv 1692 1  |-  ( A 
~<  B  ->  suc  A  ~<_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2991    \ cdif 3344    u. cun 3345    i^i cin 3346    C_ wss 3347   (/)c0 3656   {csn 3896   class class class wbr 4311   suc csuc 4740   ran crn 4860    Fn wfn 5432   -->wf 5433   -1-1->wf1 5434   -1-1-onto->wf1o 5436    ~~ cen 7326    ~<_ cdom 7327    ~< csdm 7328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-id 4655  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-1o 6939  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332
This theorem is referenced by:  sucdom  7527  card2inf  7789
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