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Theorem sucxpdom 7783
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 5444 . 2  |-  suc  A  =  ( A  u.  { A } )
2 relsdom 7580 . . . . . . . . 9  |-  Rel  ~<
32brrelex2i 4891 . . . . . . . 8  |-  ( 1o 
~<  A  ->  A  e. 
_V )
4 1on 7193 . . . . . . . 8  |-  1o  e.  On
5 xpsneng 7659 . . . . . . . 8  |-  ( ( A  e.  _V  /\  1o  e.  On )  -> 
( A  X.  { 1o } )  ~~  A
)
63, 4, 5sylancl 666 . . . . . . 7  |-  ( 1o 
~<  A  ->  ( A  X.  { 1o }
)  ~~  A )
76ensymd 7623 . . . . . 6  |-  ( 1o 
~<  A  ->  A  ~~  ( A  X.  { 1o } ) )
8 endom 7599 . . . . . 6  |-  ( A 
~~  ( A  X.  { 1o } )  ->  A  ~<_  ( A  X.  { 1o } ) )
97, 8syl 17 . . . . 5  |-  ( 1o 
~<  A  ->  A  ~<_  ( A  X.  { 1o } ) )
10 ensn1g 7637 . . . . . . . . 9  |-  ( A  e.  _V  ->  { A }  ~~  1o )
113, 10syl 17 . . . . . . . 8  |-  ( 1o 
~<  A  ->  { A }  ~~  1o )
12 ensdomtr 7710 . . . . . . . 8  |-  ( ( { A }  ~~  1o  /\  1o  ~<  A )  ->  { A }  ~<  A )
1311, 12mpancom 673 . . . . . . 7  |-  ( 1o 
~<  A  ->  { A }  ~<  A )
14 0ex 4552 . . . . . . . . 9  |-  (/)  e.  _V
15 xpsneng 7659 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
163, 14, 15sylancl 666 . . . . . . . 8  |-  ( 1o 
~<  A  ->  ( A  X.  { (/) } ) 
~~  A )
1716ensymd 7623 . . . . . . 7  |-  ( 1o 
~<  A  ->  A  ~~  ( A  X.  { (/) } ) )
18 sdomentr 7708 . . . . . . 7  |-  ( ( { A }  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  { A }  ~<  ( A  X.  { (/) } ) )
1913, 17, 18syl2anc 665 . . . . . 6  |-  ( 1o 
~<  A  ->  { A }  ~<  ( A  X.  { (/) } ) )
20 sdomdom 7600 . . . . . 6  |-  ( { A }  ~<  ( A  X.  { (/) } )  ->  { A }  ~<_  ( A  X.  { (/) } ) )
2119, 20syl 17 . . . . 5  |-  ( 1o 
~<  A  ->  { A }  ~<_  ( A  X.  { (/) } ) )
22 1n0 7201 . . . . . 6  |-  1o  =/=  (/)
23 xpsndisj 5275 . . . . . 6  |-  ( 1o  =/=  (/)  ->  ( ( A  X.  { 1o }
)  i^i  ( A  X.  { (/) } ) )  =  (/) )
2422, 23mp1i 13 . . . . 5  |-  ( 1o 
~<  A  ->  ( ( A  X.  { 1o } )  i^i  ( A  X.  { (/) } ) )  =  (/) )
25 undom 7662 . . . . 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 1263 . . . 4  |-  ( 1o 
~<  A  ->  ( A  u.  { A }
)  ~<_  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) ) )
27 sdomentr 7708 . . . . . 6  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { 1o } ) )  ->  1o  ~<  ( A  X.  { 1o }
) )
287, 27mpdan 672 . . . . 5  |-  ( 1o 
~<  A  ->  1o  ~<  ( A  X.  { 1o } ) )
29 sdomentr 7708 . . . . . 6  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  1o  ~<  ( A  X.  { (/) } ) )
3017, 29mpdan 672 . . . . 5  |-  ( 1o 
~<  A  ->  1o  ~<  ( A  X.  { (/) } ) )
31 unxpdom 7781 . . . . 5  |-  ( ( 1o  ~<  ( A  X.  { 1o } )  /\  1o  ~<  ( A  X.  { (/) } ) )  ->  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) )  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )
3228, 30, 31syl2anc 665 . . . 4  |-  ( 1o 
~<  A  ->  ( ( A  X.  { 1o } )  u.  ( A  X.  { (/) } ) )  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )
33 domtr 7625 . . . 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 665 . . 3  |-  ( 1o 
~<  A  ->  ( A  u.  { A }
)  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )
35 xpen 7737 . . . 4  |-  ( ( ( A  X.  { 1o } )  ~~  A  /\  ( A  X.  { (/)
} )  ~~  A
)  ->  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) )  ~~  ( A  X.  A ) )
366, 16, 35syl2anc 665 . . 3  |-  ( 1o 
~<  A  ->  ( ( A  X.  { 1o } )  X.  ( A  X.  { (/) } ) )  ~~  ( A  X.  A ) )
37 domentr 7631 . . 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 665 . 2  |-  ( 1o 
~<  A  ->  ( A  u.  { A }
)  ~<_  ( A  X.  A ) )
391, 38syl5eqbr 4454 1  |-  ( 1o 
~<  A  ->  suc  A  ~<_  ( A  X.  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1868    =/= wne 2618   _Vcvv 3081    u. cun 3434    i^i cin 3435   (/)c0 3761   {csn 3996   class class class wbr 4420    X. cxp 4847   Oncon0 5438   suc csuc 5440   1oc1o 7179    ~~ cen 7570    ~<_ cdom 7571    ~< csdm 7572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-om 6703  df-1st 6803  df-2nd 6804  df-1o 7186  df-2o 7187  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576
This theorem is referenced by: (None)
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