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Theorem sucxpdom 7514
Description: Cartesian 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 4720 . 2  |-  suc  A  =  ( A  u.  { A } )
2 relsdom 7309 . . . . . . . . 9  |-  Rel  ~<
32brrelex2i 4875 . . . . . . . 8  |-  ( 1o 
~<  A  ->  A  e. 
_V )
4 1on 6919 . . . . . . . 8  |-  1o  e.  On
5 xpsneng 7388 . . . . . . . 8  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  X.  { 1o } )  ~~  A
)
63, 4, 5sylancl 662 . . . . . . 7  |-  ( 1o 
~<  A  ->  ( A  X.  { 1o }
)  ~~  A )
76ensymd 7352 . . . . . 6  |-  ( 1o 
~<  A  ->  A  ~~  ( A  X.  { 1o } ) )
8 endom 7328 . . . . . 6  |-  ( A 
~~  ( A  X.  { 1o } )  ->  A  ~<_  ( A  X.  { 1o } ) )
97, 8syl 16 . . . . 5  |-  ( 1o 
~<  A  ->  A  ~<_  ( A  X.  { 1o } ) )
10 ensn1g 7366 . . . . . . . . 9  |-  ( A  e.  _V  ->  { A }  ~~  1o )
113, 10syl 16 . . . . . . . 8  |-  ( 1o 
~<  A  ->  { A }  ~~  1o )
12 ensdomtr 7439 . . . . . . . 8  |-  ( ( { A }  ~~  1o  /\  1o  ~<  A )  ->  { A }  ~<  A )
1311, 12mpancom 669 . . . . . . 7  |-  ( 1o 
~<  A  ->  { A }  ~<  A )
14 0ex 4417 . . . . . . . . 9  |-  (/)  e.  _V
15 xpsneng 7388 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
163, 14, 15sylancl 662 . . . . . . . 8  |-  ( 1o 
~<  A  ->  ( A  X.  { (/) } ) 
~~  A )
1716ensymd 7352 . . . . . . 7  |-  ( 1o 
~<  A  ->  A  ~~  ( A  X.  { (/) } ) )
18 sdomentr 7437 . . . . . . 7  |-  ( ( { A }  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  { A }  ~<  ( A  X.  { (/) } ) )
1913, 17, 18syl2anc 661 . . . . . 6  |-  ( 1o 
~<  A  ->  { A }  ~<  ( A  X.  { (/) } ) )
20 sdomdom 7329 . . . . . 6  |-  ( { A }  ~<  ( A  X.  { (/) } )  ->  { A }  ~<_  ( A  X.  { (/) } ) )
2119, 20syl 16 . . . . 5  |-  ( 1o 
~<  A  ->  { A }  ~<_  ( A  X.  { (/) } ) )
22 1n0 6927 . . . . . 6  |-  1o  =/=  (/)
23 xpsndisj 5256 . . . . . 6  |-  ( 1o  =/=  (/)  ->  ( ( A  X.  { 1o }
)  i^i  ( A  X.  { (/) } ) )  =  (/) )
2422, 23mp1i 12 . . . . 5  |-  ( 1o 
~<  A  ->  ( ( A  X.  { 1o } )  i^i  ( A  X.  { (/) } ) )  =  (/) )
25 undom 7391 . . . . 5  |-  ( ( ( A  ~<_  ( A  X.  { 1o }
)  /\  { A }  ~<_  ( A  X.  { (/) } ) )  /\  ( ( A  X.  { 1o }
)  i^i  ( A  X.  { (/) } ) )  =  (/) )  ->  ( A  u.  { A } )  ~<_  ( ( A  X.  { 1o } )  u.  ( A  X.  { (/) } ) ) )
269, 21, 24, 25syl21anc 1217 . . . 4  |-  ( 1o 
~<  A  ->  ( A  u.  { A }
)  ~<_  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) ) )
27 sdomentr 7437 . . . . . 6  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { 1o } ) )  ->  1o  ~<  ( A  X.  { 1o }
) )
287, 27mpdan 668 . . . . 5  |-  ( 1o 
~<  A  ->  1o  ~<  ( A  X.  { 1o } ) )
29 sdomentr 7437 . . . . . 6  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  1o  ~<  ( A  X.  { (/) } ) )
3017, 29mpdan 668 . . . . 5  |-  ( 1o 
~<  A  ->  1o  ~<  ( A  X.  { (/) } ) )
31 unxpdom 7512 . . . . 5  |-  ( ( 1o  ~<  ( A  X.  { 1o } )  /\  1o  ~<  ( A  X.  { (/) } ) )  ->  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) )  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )
3228, 30, 31syl2anc 661 . . . 4  |-  ( 1o 
~<  A  ->  ( ( A  X.  { 1o } )  u.  ( A  X.  { (/) } ) )  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )
33 domtr 7354 . . . 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.  { (/) } ) ) )
3426, 32, 33syl2anc 661 . . 3  |-  ( 1o 
~<  A  ->  ( A  u.  { A }
)  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )
35 xpen 7466 . . . 4  |-  ( ( ( A  X.  { 1o } )  ~~  A  /\  ( A  X.  { (/)
} )  ~~  A
)  ->  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) )  ~~  ( A  X.  A ) )
366, 16, 35syl2anc 661 . . 3  |-  ( 1o 
~<  A  ->  ( ( A  X.  { 1o } )  X.  ( A  X.  { (/) } ) )  ~~  ( A  X.  A ) )
37 domentr 7360 . . 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 ) )
3834, 36, 37syl2anc 661 . 2  |-  ( 1o 
~<  A  ->  ( A  u.  { A }
)  ~<_  ( A  X.  A ) )
391, 38syl5eqbr 4320 1  |-  ( 1o 
~<  A  ->  suc  A  ~<_  ( A  X.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    =/= wne 2601   _Vcvv 2967    u. cun 3321    i^i cin 3322   (/)c0 3632   {csn 3872   class class class wbr 4287   Oncon0 4714   suc csuc 4716    X. cxp 4833   1oc1o 6905    ~~ cen 7299    ~<_ cdom 7300    ~< csdm 7301
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-1st 6572  df-2nd 6573  df-1o 6912  df-2o 6913  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305
This theorem is referenced by: (None)
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