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Theorem cdaxpdom 7699
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 6756 . . . . 5  |-  Rel  ~<
21brrelex2i 4637 . . . 4  |-  ( 1o 
~<  A  ->  A  e. 
_V )
31brrelex2i 4637 . . . 4  |-  ( 1o 
~<  B  ->  B  e. 
_V )
4 cdaval 7680 . . . 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 4047 . . . . . . 7  |-  (/)  e.  _V
7 xpsneng 6832 . . . . . . 7  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
82, 6, 7sylancl 646 . . . . . 6  |-  ( 1o 
~<  A  ->  ( A  X.  { (/) } ) 
~~  A )
9 sdomen2 6891 . . . . . 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 6372 . . . . . . 7  |-  1o  e.  On
13 xpsneng 6832 . . . . . . 7  |-  ( ( B  e.  _V  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
143, 12, 13sylancl 646 . . . . . 6  |-  ( 1o 
~<  B  ->  ( B  X.  { 1o }
)  ~~  B )
15 sdomen2 6891 . . . . . 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 6955 . . . 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 3940 . 2  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  +c  B
)  ~<_  ( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) ) )
21 xpen 6909 . . 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 6805 . 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 2727    u. cun 3076   (/)c0 3362   {csn 3544   class class class wbr 3920   Oncon0 4285    X. cxp 4578  (class class class)co 5710   1oc1o 6358    ~~ cen 6746    ~<_ cdom 6747    ~< csdm 6748    +c ccda 7677
This theorem is referenced by:  canthp1lem1  8154
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-1o 6365  df-2o 6366  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-cda 7678
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