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Theorem cdaxpdom 8002
Description: Cross product dominates disjoint union for sets with cardinality greater than 1. Similar to Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
cdaxpdom  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  +c  B
)  ~<_  ( A  X.  B ) )

Proof of Theorem cdaxpdom
StepHypRef Expression
1 relsdom 7052 . . . . 5  |-  Rel  ~<
21brrelex2i 4859 . . . 4  |-  ( 1o 
~<  A  ->  A  e. 
_V )
31brrelex2i 4859 . . . 4  |-  ( 1o 
~<  B  ->  B  e. 
_V )
4 cdaval 7983 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
52, 3, 4syl2an 464 . . 3  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
6 0ex 4280 . . . . . . 7  |-  (/)  e.  _V
7 xpsneng 7129 . . . . . . 7  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
82, 6, 7sylancl 644 . . . . . 6  |-  ( 1o 
~<  A  ->  ( A  X.  { (/) } ) 
~~  A )
9 sdomen2 7188 . . . . . 6  |-  ( ( A  X.  { (/) } )  ~~  A  -> 
( 1o  ~<  ( A  X.  { (/) } )  <-> 
1o  ~<  A ) )
108, 9syl 16 . . . . 5  |-  ( 1o 
~<  A  ->  ( 1o 
~<  ( A  X.  { (/)
} )  <->  1o  ~<  A ) )
1110ibir 234 . . . 4  |-  ( 1o 
~<  A  ->  1o  ~<  ( A  X.  { (/) } ) )
12 1on 6667 . . . . . . 7  |-  1o  e.  On
13 xpsneng 7129 . . . . . . 7  |-  ( ( B  e.  _V  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
143, 12, 13sylancl 644 . . . . . 6  |-  ( 1o 
~<  B  ->  ( B  X.  { 1o }
)  ~~  B )
15 sdomen2 7188 . . . . . 6  |-  ( ( B  X.  { 1o } )  ~~  B  ->  ( 1o  ~<  ( B  X.  { 1o }
)  <->  1o  ~<  B ) )
1614, 15syl 16 . . . . 5  |-  ( 1o 
~<  B  ->  ( 1o 
~<  ( B  X.  { 1o } )  <->  1o  ~<  B ) )
1716ibir 234 . . . 4  |-  ( 1o 
~<  B  ->  1o  ~<  ( B  X.  { 1o } ) )
18 unxpdom 7252 . . . 4  |-  ( ( 1o  ~<  ( A  X.  { (/) } )  /\  1o  ~<  ( B  X.  { 1o } ) )  ->  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) )  ~<_  ( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) ) )
1911, 17, 18syl2an 464 . . 3  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) )  ~<_  ( ( A  X.  { (/)
} )  X.  ( B  X.  { 1o }
) ) )
205, 19eqbrtrd 4173 . 2  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  +c  B
)  ~<_  ( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) ) )
21 xpen 7206 . . 3  |-  ( ( ( A  X.  { (/)
} )  ~~  A  /\  ( B  X.  { 1o } )  ~~  B
)  ->  ( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) ) 
~~  ( A  X.  B ) )
228, 14, 21syl2an 464 . 2  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) )  ~~  ( A  X.  B
) )
23 domentr 7102 . 2  |-  ( ( ( A  +c  B
)  ~<_  ( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) )  /\  ( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) ) 
~~  ( A  X.  B ) )  -> 
( A  +c  B
)  ~<_  ( A  X.  B ) )
2420, 22, 23syl2anc 643 1  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  +c  B
)  ~<_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899    u. cun 3261   (/)c0 3571   {csn 3757   class class class wbr 4153   Oncon0 4522    X. cxp 4816  (class class class)co 6020   1oc1o 6653    ~~ cen 7042    ~<_ cdom 7043    ~< csdm 7044    +c ccda 7980
This theorem is referenced by:  canthp1lem1  8460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-1o 6660  df-2o 6661  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-cda 7981
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