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Theorem cdaxpdom 7748
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 6803 . . . . 5  |-  Rel  ~<
21brrelex2i 4683 . . . 4  |-  ( 1o 
~<  A  ->  A  e. 
_V )
31brrelex2i 4683 . . . 4  |-  ( 1o 
~<  B  ->  B  e. 
_V )
4 cdaval 7729 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
52, 3, 4syl2an 465 . . 3  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
6 0ex 4090 . . . . . . 7  |-  (/)  e.  _V
7 xpsneng 6880 . . . . . . 7  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
82, 6, 7sylancl 646 . . . . . 6  |-  ( 1o 
~<  A  ->  ( A  X.  { (/) } ) 
~~  A )
9 sdomen2 6939 . . . . . 6  |-  ( ( A  X.  { (/) } )  ~~  A  -> 
( 1o  ~<  ( A  X.  { (/) } )  <-> 
1o  ~<  A ) )
108, 9syl 17 . . . . 5  |-  ( 1o 
~<  A  ->  ( 1o 
~<  ( A  X.  { (/)
} )  <->  1o  ~<  A ) )
1110ibir 235 . . . 4  |-  ( 1o 
~<  A  ->  1o  ~<  ( A  X.  { (/) } ) )
12 1on 6419 . . . . . . 7  |-  1o  e.  On
13 xpsneng 6880 . . . . . . 7  |-  ( ( B  e.  _V  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
143, 12, 13sylancl 646 . . . . . 6  |-  ( 1o 
~<  B  ->  ( B  X.  { 1o }
)  ~~  B )
15 sdomen2 6939 . . . . . 6  |-  ( ( B  X.  { 1o } )  ~~  B  ->  ( 1o  ~<  ( B  X.  { 1o }
)  <->  1o  ~<  B ) )
1614, 15syl 17 . . . . 5  |-  ( 1o 
~<  B  ->  ( 1o 
~<  ( B  X.  { 1o } )  <->  1o  ~<  B ) )
1716ibir 235 . . . 4  |-  ( 1o 
~<  B  ->  1o  ~<  ( B  X.  { 1o } ) )
18 unxpdom 7003 . . . 4  |-  ( ( 1o  ~<  ( A  X.  { (/) } )  /\  1o  ~<  ( B  X.  { 1o } ) )  ->  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) )  ~<_  ( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) ) )
1911, 17, 18syl2an 465 . . 3  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) )  ~<_  ( ( A  X.  { (/)
} )  X.  ( B  X.  { 1o }
) ) )
205, 19eqbrtrd 3983 . 2  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  +c  B
)  ~<_  ( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) ) )
21 xpen 6957 . . 3  |-  ( ( ( A  X.  { (/)
} )  ~~  A  /\  ( B  X.  { 1o } )  ~~  B
)  ->  ( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) ) 
~~  ( A  X.  B ) )
228, 14, 21syl2an 465 . 2  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) )  ~~  ( A  X.  B
) )
23 domentr 6853 . 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 645 1  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  +c  B
)  ~<_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2740    u. cun 3092   (/)c0 3397   {csn 3581   class class class wbr 3963   Oncon0 4329    X. cxp 4624  (class class class)co 5757   1oc1o 6405    ~~ cen 6793    ~<_ cdom 6794    ~< csdm 6795    +c ccda 7726
This theorem is referenced by:  canthp1lem1  8207
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-1o 6412  df-2o 6413  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-cda 7727
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