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Theorem cdaxpdom 8025
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 7075 . . . . 5  |-  Rel  ~<
21brrelex2i 4878 . . . 4  |-  ( 1o 
~<  A  ->  A  e. 
_V )
31brrelex2i 4878 . . . 4  |-  ( 1o 
~<  B  ->  B  e. 
_V )
4 cdaval 8006 . . . 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 4299 . . . . . . 7  |-  (/)  e.  _V
7 xpsneng 7152 . . . . . . 7  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
82, 6, 7sylancl 644 . . . . . 6  |-  ( 1o 
~<  A  ->  ( A  X.  { (/) } ) 
~~  A )
9 sdomen2 7211 . . . . . 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 6690 . . . . . . 7  |-  1o  e.  On
13 xpsneng 7152 . . . . . . 7  |-  ( ( B  e.  _V  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
143, 12, 13sylancl 644 . . . . . 6  |-  ( 1o 
~<  B  ->  ( B  X.  { 1o }
)  ~~  B )
15 sdomen2 7211 . . . . . 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 7275 . . . 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 4192 . 2  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  +c  B
)  ~<_  ( ( A  X.  { (/) } )  X.  ( B  X.  { 1o } ) ) )
21 xpen 7229 . . 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 7125 . 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 2916    u. cun 3278   (/)c0 3588   {csn 3774   class class class wbr 4172   Oncon0 4541    X. cxp 4835  (class class class)co 6040   1oc1o 6676    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067    +c ccda 8003
This theorem is referenced by:  canthp1lem1  8483
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-1o 6683  df-2o 6684  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-cda 8004
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