MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sucxpdom Unicode version

Theorem sucxpdom 7026
Description: Cross product dominates successor for set with cardinality greater than 1. Proposition 10.38 of [TakeutiZaring] p. 93 (but generalized to arbitrary sets, not just ordinals). (Contributed by NM, 3-Sep-2004.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
sucxpdom  |-  ( 1o 
~<  A  ->  suc  A  ~<_  ( A  X.  A
) )

Proof of Theorem sucxpdom
StepHypRef Expression
1 df-suc 4356 . 2  |-  suc  A  =  ( A  u.  { A } )
2 relsdom 6824 . . . . . . . . 9  |-  Rel  ~<
32brrelex2i 4704 . . . . . . . 8  |-  ( 1o 
~<  A  ->  A  e. 
_V )
4 1on 6440 . . . . . . . 8  |-  1o  e.  On
5 xpsneng 6901 . . . . . . . 8  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  X.  { 1o } )  ~~  A
)
63, 4, 5sylancl 646 . . . . . . 7  |-  ( 1o 
~<  A  ->  ( A  X.  { 1o }
)  ~~  A )
7 ensym 6864 . . . . . . 7  |-  ( ( A  X.  { 1o } )  ~~  A  ->  A  ~~  ( A  X.  { 1o }
) )
86, 7syl 17 . . . . . 6  |-  ( 1o 
~<  A  ->  A  ~~  ( A  X.  { 1o } ) )
9 endom 6842 . . . . . 6  |-  ( A 
~~  ( A  X.  { 1o } )  ->  A  ~<_  ( A  X.  { 1o } ) )
108, 9syl 17 . . . . 5  |-  ( 1o 
~<  A  ->  A  ~<_  ( A  X.  { 1o } ) )
11 ensn1g 6880 . . . . . . . . 9  |-  ( A  e.  _V  ->  { A }  ~~  1o )
123, 11syl 17 . . . . . . . 8  |-  ( 1o 
~<  A  ->  { A }  ~~  1o )
13 ensdomtr 6951 . . . . . . . 8  |-  ( ( { A }  ~~  1o  /\  1o  ~<  A )  ->  { A }  ~<  A )
1412, 13mpancom 653 . . . . . . 7  |-  ( 1o 
~<  A  ->  { A }  ~<  A )
15 0ex 4110 . . . . . . . . 9  |-  (/)  e.  _V
16 xpsneng 6901 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
173, 15, 16sylancl 646 . . . . . . . 8  |-  ( 1o 
~<  A  ->  ( A  X.  { (/) } ) 
~~  A )
18 ensym 6864 . . . . . . . 8  |-  ( ( A  X.  { (/) } )  ~~  A  ->  A  ~~  ( A  X.  { (/) } ) )
1917, 18syl 17 . . . . . . 7  |-  ( 1o 
~<  A  ->  A  ~~  ( A  X.  { (/) } ) )
20 sdomentr 6949 . . . . . . 7  |-  ( ( { A }  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  { A }  ~<  ( A  X.  { (/) } ) )
2114, 19, 20syl2anc 645 . . . . . 6  |-  ( 1o 
~<  A  ->  { A }  ~<  ( A  X.  { (/) } ) )
22 sdomdom 6843 . . . . . 6  |-  ( { A }  ~<  ( A  X.  { (/) } )  ->  { A }  ~<_  ( A  X.  { (/) } ) )
2321, 22syl 17 . . . . 5  |-  ( 1o 
~<  A  ->  { A }  ~<_  ( A  X.  { (/) } ) )
24 1n0 6448 . . . . . 6  |-  1o  =/=  (/)
25 xpsndisj 5077 . . . . . 6  |-  ( 1o  =/=  (/)  ->  ( ( A  X.  { 1o }
)  i^i  ( A  X.  { (/) } ) )  =  (/) )
2624, 25mp1i 13 . . . . 5  |-  ( 1o 
~<  A  ->  ( ( A  X.  { 1o } )  i^i  ( A  X.  { (/) } ) )  =  (/) )
27 undom 6904 . . . . 5  |-  ( ( ( A  ~<_  ( A  X.  { 1o }
)  /\  { A }  ~<_  ( A  X.  { (/) } ) )  /\  ( ( A  X.  { 1o }
)  i^i  ( A  X.  { (/) } ) )  =  (/) )  ->  ( A  u.  { A } )  ~<_  ( ( A  X.  { 1o } )  u.  ( A  X.  { (/) } ) ) )
2810, 23, 26, 27syl21anc 1186 . . . 4  |-  ( 1o 
~<  A  ->  ( A  u.  { A }
)  ~<_  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) ) )
29 sdomentr 6949 . . . . . 6  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { 1o } ) )  ->  1o  ~<  ( A  X.  { 1o }
) )
308, 29mpdan 652 . . . . 5  |-  ( 1o 
~<  A  ->  1o  ~<  ( A  X.  { 1o } ) )
31 sdomentr 6949 . . . . . 6  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  1o  ~<  ( A  X.  { (/) } ) )
3219, 31mpdan 652 . . . . 5  |-  ( 1o 
~<  A  ->  1o  ~<  ( A  X.  { (/) } ) )
33 unxpdom 7024 . . . . 5  |-  ( ( 1o  ~<  ( A  X.  { 1o } )  /\  1o  ~<  ( A  X.  { (/) } ) )  ->  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) )  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )
3430, 32, 33syl2anc 645 . . . 4  |-  ( 1o 
~<  A  ->  ( ( A  X.  { 1o } )  u.  ( A  X.  { (/) } ) )  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )
35 domtr 6868 . . . 4  |-  ( ( ( A  u.  { A } )  ~<_  ( ( A  X.  { 1o } )  u.  ( A  X.  { (/) } ) )  /\  ( ( A  X.  { 1o } )  u.  ( A  X.  { (/) } ) )  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )  ->  ( A  u.  { A } )  ~<_  ( ( A  X.  { 1o } )  X.  ( A  X.  { (/) } ) ) )
3628, 34, 35syl2anc 645 . . 3  |-  ( 1o 
~<  A  ->  ( A  u.  { A }
)  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )
37 xpen 6978 . . . 4  |-  ( ( ( A  X.  { 1o } )  ~~  A  /\  ( A  X.  { (/)
} )  ~~  A
)  ->  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) )  ~~  ( A  X.  A ) )
386, 17, 37syl2anc 645 . . 3  |-  ( 1o 
~<  A  ->  ( ( A  X.  { 1o } )  X.  ( A  X.  { (/) } ) )  ~~  ( A  X.  A ) )
39 domentr 6874 . . 3  |-  ( ( ( A  u.  { A } )  ~<_  ( ( A  X.  { 1o } )  X.  ( A  X.  { (/) } ) )  /\  ( ( A  X.  { 1o } )  X.  ( A  X.  { (/) } ) )  ~~  ( A  X.  A ) )  ->  ( A  u.  { A } )  ~<_  ( A  X.  A ) )
4036, 38, 39syl2anc 645 . 2  |-  ( 1o 
~<  A  ->  ( A  u.  { A }
)  ~<_  ( A  X.  A ) )
411, 40syl5eqbr 4016 1  |-  ( 1o 
~<  A  ->  suc  A  ~<_  ( A  X.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621    =/= wne 2419   _Vcvv 2757    u. cun 3111    i^i cin 3112   (/)c0 3416   {csn 3600   class class class wbr 3983   Oncon0 4350   suc csuc 4352    X. cxp 4645   1oc1o 6426    ~~ cen 6814    ~<_ cdom 6815    ~< csdm 6816
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-1st 6042  df-2nd 6043  df-1o 6433  df-2o 6434  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820
  Copyright terms: Public domain W3C validator