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Theorem sucxpdom 6957
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 4291 . 2  |-  suc  A  =  ( A  u.  { A } )
2 relsdom 6756 . . . . . . . . 9  |-  Rel  ~<
32brrelex2i 4637 . . . . . . . 8  |-  ( 1o 
~<  A  ->  A  e. 
_V )
4 1on 6372 . . . . . . . 8  |-  1o  e.  On
5 xpsneng 6832 . . . . . . . 8  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  X.  { 1o } )  ~~  A
)
63, 4, 5sylancl 646 . . . . . . 7  |-  ( 1o 
~<  A  ->  ( A  X.  { 1o }
)  ~~  A )
7 ensym 6796 . . . . . . 7  |-  ( ( A  X.  { 1o } )  ~~  A  ->  A  ~~  ( A  X.  { 1o }
) )
86, 7syl 17 . . . . . 6  |-  ( 1o 
~<  A  ->  A  ~~  ( A  X.  { 1o } ) )
9 endom 6774 . . . . . 6  |-  ( A 
~~  ( A  X.  { 1o } )  ->  A  ~<_  ( A  X.  { 1o } ) )
108, 9syl 17 . . . . 5  |-  ( 1o 
~<  A  ->  A  ~<_  ( A  X.  { 1o } ) )
11 ensn1g 6811 . . . . . . . . 9  |-  ( A  e.  _V  ->  { A }  ~~  1o )
123, 11syl 17 . . . . . . . 8  |-  ( 1o 
~<  A  ->  { A }  ~~  1o )
13 ensdomtr 6882 . . . . . . . 8  |-  ( ( { A }  ~~  1o  /\  1o  ~<  A )  ->  { A }  ~<  A )
1412, 13mpancom 653 . . . . . . 7  |-  ( 1o 
~<  A  ->  { A }  ~<  A )
15 0ex 4047 . . . . . . . . 9  |-  (/)  e.  _V
16 xpsneng 6832 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
173, 15, 16sylancl 646 . . . . . . . 8  |-  ( 1o 
~<  A  ->  ( A  X.  { (/) } ) 
~~  A )
18 ensym 6796 . . . . . . . 8  |-  ( ( A  X.  { (/) } )  ~~  A  ->  A  ~~  ( A  X.  { (/) } ) )
1917, 18syl 17 . . . . . . 7  |-  ( 1o 
~<  A  ->  A  ~~  ( A  X.  { (/) } ) )
20 sdomentr 6880 . . . . . . 7  |-  ( ( { A }  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  { A }  ~<  ( A  X.  { (/) } ) )
2114, 19, 20syl2anc 645 . . . . . 6  |-  ( 1o 
~<  A  ->  { A }  ~<  ( A  X.  { (/) } ) )
22 sdomdom 6775 . . . . . 6  |-  ( { A }  ~<  ( A  X.  { (/) } )  ->  { A }  ~<_  ( A  X.  { (/) } ) )
2321, 22syl 17 . . . . 5  |-  ( 1o 
~<  A  ->  { A }  ~<_  ( A  X.  { (/) } ) )
24 1n0 6380 . . . . . 6  |-  1o  =/=  (/)
25 xpsndisj 5010 . . . . . 6  |-  ( 1o  =/=  (/)  ->  ( ( A  X.  { 1o }
)  i^i  ( A  X.  { (/) } ) )  =  (/) )
2624, 25mp1i 13 . . . . 5  |-  ( 1o 
~<  A  ->  ( ( A  X.  { 1o } )  i^i  ( A  X.  { (/) } ) )  =  (/) )
27 undom 6835 . . . . 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 6880 . . . . . 6  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { 1o } ) )  ->  1o  ~<  ( A  X.  { 1o }
) )
308, 29mpdan 652 . . . . 5  |-  ( 1o 
~<  A  ->  1o  ~<  ( A  X.  { 1o } ) )
31 sdomentr 6880 . . . . . 6  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  1o  ~<  ( A  X.  { (/) } ) )
3219, 31mpdan 652 . . . . 5  |-  ( 1o 
~<  A  ->  1o  ~<  ( A  X.  { (/) } ) )
33 unxpdom 6955 . . . . 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 6799 . . . 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 6909 . . . 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 6805 . . 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 3953 1  |-  ( 1o 
~<  A  ->  suc  A  ~<_  ( A  X.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621    =/= wne 2412   _Vcvv 2727    u. cun 3076    i^i cin 3077   (/)c0 3362   {csn 3544   class class class wbr 3920   Oncon0 4285   suc csuc 4287    X. cxp 4578   1oc1o 6358    ~~ cen 6746    ~<_ cdom 6747    ~< csdm 6748
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-1st 5974  df-2nd 5975  df-1o 6365  df-2o 6366  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752
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