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Theorem cdaxpdom 8033
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 7083 . . . . 5  |-  Rel  ~<
21brrelex2i 4886 . . . 4  |-  ( 1o 
~<  A  ->  A  e. 
_V )
31brrelex2i 4886 . . . 4  |-  ( 1o 
~<  B  ->  B  e. 
_V )
4 cdaval 8014 . . . 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 4307 . . . . . . 7  |-  (/)  e.  _V
7 xpsneng 7160 . . . . . . 7  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
82, 6, 7sylancl 644 . . . . . 6  |-  ( 1o 
~<  A  ->  ( A  X.  { (/) } ) 
~~  A )
9 sdomen2 7219 . . . . . 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 6698 . . . . . . 7  |-  1o  e.  On
13 xpsneng 7160 . . . . . . 7  |-  ( ( B  e.  _V  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
143, 12, 13sylancl 644 . . . . . 6  |-  ( 1o 
~<  B  ->  ( B  X.  { 1o }
)  ~~  B )
15 sdomen2 7219 . . . . . 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 7283 . . . 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 4200 . 2  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  +c  B
)  ~<_  ( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) ) )
21 xpen 7237 . . 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 7133 . 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 1721   _Vcvv 2924    u. cun 3286   (/)c0 3596   {csn 3782   class class class wbr 4180   Oncon0 4549    X. cxp 4843  (class class class)co 6048   1oc1o 6684    ~~ cen 7073    ~<_ cdom 7074    ~< csdm 7075    +c ccda 8011
This theorem is referenced by:  canthp1lem1  8491
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-1o 6691  df-2o 6692  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-cda 8012
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