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Theorem sucxpdom 7748
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 4893 . 2  |-  suc  A  =  ( A  u.  { A } )
2 relsdom 7542 . . . . . . . . 9  |-  Rel  ~<
32brrelex2i 5050 . . . . . . . 8  |-  ( 1o 
~<  A  ->  A  e. 
_V )
4 1on 7155 . . . . . . . 8  |-  1o  e.  On
5 xpsneng 7621 . . . . . . . 8  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  X.  { 1o } )  ~~  A
)
63, 4, 5sylancl 662 . . . . . . 7  |-  ( 1o 
~<  A  ->  ( A  X.  { 1o }
)  ~~  A )
76ensymd 7585 . . . . . 6  |-  ( 1o 
~<  A  ->  A  ~~  ( A  X.  { 1o } ) )
8 endom 7561 . . . . . 6  |-  ( A 
~~  ( A  X.  { 1o } )  ->  A  ~<_  ( A  X.  { 1o } ) )
97, 8syl 16 . . . . 5  |-  ( 1o 
~<  A  ->  A  ~<_  ( A  X.  { 1o } ) )
10 ensn1g 7599 . . . . . . . . 9  |-  ( A  e.  _V  ->  { A }  ~~  1o )
113, 10syl 16 . . . . . . . 8  |-  ( 1o 
~<  A  ->  { A }  ~~  1o )
12 ensdomtr 7672 . . . . . . . 8  |-  ( ( { A }  ~~  1o  /\  1o  ~<  A )  ->  { A }  ~<  A )
1311, 12mpancom 669 . . . . . . 7  |-  ( 1o 
~<  A  ->  { A }  ~<  A )
14 0ex 4587 . . . . . . . . 9  |-  (/)  e.  _V
15 xpsneng 7621 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
163, 14, 15sylancl 662 . . . . . . . 8  |-  ( 1o 
~<  A  ->  ( A  X.  { (/) } ) 
~~  A )
1716ensymd 7585 . . . . . . 7  |-  ( 1o 
~<  A  ->  A  ~~  ( A  X.  { (/) } ) )
18 sdomentr 7670 . . . . . . 7  |-  ( ( { A }  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  { A }  ~<  ( A  X.  { (/) } ) )
1913, 17, 18syl2anc 661 . . . . . 6  |-  ( 1o 
~<  A  ->  { A }  ~<  ( A  X.  { (/) } ) )
20 sdomdom 7562 . . . . . 6  |-  ( { A }  ~<  ( A  X.  { (/) } )  ->  { A }  ~<_  ( A  X.  { (/) } ) )
2119, 20syl 16 . . . . 5  |-  ( 1o 
~<  A  ->  { A }  ~<_  ( A  X.  { (/) } ) )
22 1n0 7163 . . . . . 6  |-  1o  =/=  (/)
23 xpsndisj 5437 . . . . . 6  |-  ( 1o  =/=  (/)  ->  ( ( A  X.  { 1o }
)  i^i  ( A  X.  { (/) } ) )  =  (/) )
2422, 23mp1i 12 . . . . 5  |-  ( 1o 
~<  A  ->  ( ( A  X.  { 1o } )  i^i  ( A  X.  { (/) } ) )  =  (/) )
25 undom 7624 . . . . 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 1227 . . . 4  |-  ( 1o 
~<  A  ->  ( A  u.  { A }
)  ~<_  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) ) )
27 sdomentr 7670 . . . . . 6  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { 1o } ) )  ->  1o  ~<  ( A  X.  { 1o }
) )
287, 27mpdan 668 . . . . 5  |-  ( 1o 
~<  A  ->  1o  ~<  ( A  X.  { 1o } ) )
29 sdomentr 7670 . . . . . 6  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  1o  ~<  ( A  X.  { (/) } ) )
3017, 29mpdan 668 . . . . 5  |-  ( 1o 
~<  A  ->  1o  ~<  ( A  X.  { (/) } ) )
31 unxpdom 7746 . . . . 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 7587 . . . 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 7699 . . . 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 7593 . . 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 4489 1  |-  ( 1o 
~<  A  ->  suc  A  ~<_  ( A  X.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    u. cun 3469    i^i cin 3470   (/)c0 3793   {csn 4032   class class class wbr 4456   Oncon0 4887   suc csuc 4889    X. cxp 5006   1oc1o 7141    ~~ cen 7532    ~<_ cdom 7533    ~< csdm 7534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-1st 6799  df-2nd 6800  df-1o 7148  df-2o 7149  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538
This theorem is referenced by: (None)
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