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Theorem unxpdom 7038
Description: Cross product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 13-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
unxpdom  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  u.  B
)  ~<_  ( A  X.  B ) )

Proof of Theorem unxpdom
StepHypRef Expression
1 relsdom 6838 . . . 4  |-  Rel  ~<
21brrelex2i 4718 . . 3  |-  ( 1o 
~<  A  ->  A  e. 
_V )
31brrelex2i 4718 . . 3  |-  ( 1o 
~<  B  ->  B  e. 
_V )
42, 3anim12i 551 . 2  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  e.  _V  /\  B  e.  _V )
)
5 breq2 4001 . . . . 5  |-  ( x  =  A  ->  ( 1o  ~<  x  <->  1o  ~<  A ) )
65anbi1d 688 . . . 4  |-  ( x  =  A  ->  (
( 1o  ~<  x  /\  1o  ~<  y )  <->  ( 1o  ~<  A  /\  1o  ~<  y ) ) )
7 uneq1 3297 . . . . 5  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
8 xpeq1 4691 . . . . 5  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
97, 8breq12d 4010 . . . 4  |-  ( x  =  A  ->  (
( x  u.  y
)  ~<_  ( x  X.  y )  <->  ( A  u.  y )  ~<_  ( A  X.  y ) ) )
106, 9imbi12d 313 . . 3  |-  ( x  =  A  ->  (
( ( 1o  ~<  x  /\  1o  ~<  y
)  ->  ( x  u.  y )  ~<_  ( x  X.  y ) )  <-> 
( ( 1o  ~<  A  /\  1o  ~<  y
)  ->  ( A  u.  y )  ~<_  ( A  X.  y ) ) ) )
11 breq2 4001 . . . . 5  |-  ( y  =  B  ->  ( 1o  ~<  y  <->  1o  ~<  B ) )
1211anbi2d 687 . . . 4  |-  ( y  =  B  ->  (
( 1o  ~<  A  /\  1o  ~<  y )  <->  ( 1o  ~<  A  /\  1o  ~<  B ) ) )
13 uneq2 3298 . . . . 5  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
14 xpeq2 4692 . . . . 5  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
1513, 14breq12d 4010 . . . 4  |-  ( y  =  B  ->  (
( A  u.  y
)  ~<_  ( A  X.  y )  <->  ( A  u.  B )  ~<_  ( A  X.  B ) ) )
1612, 15imbi12d 313 . . 3  |-  ( y  =  B  ->  (
( ( 1o  ~<  A  /\  1o  ~<  y
)  ->  ( A  u.  y )  ~<_  ( A  X.  y ) )  <-> 
( ( 1o  ~<  A  /\  1o  ~<  B )  ->  ( A  u.  B )  ~<_  ( A  X.  B ) ) ) )
17 eqid 2258 . . . 4  |-  ( z  e.  ( x  u.  y )  |->  if ( z  e.  x , 
<. z ,  if ( z  =  v ,  w ,  t )
>. ,  <. if ( z  =  w ,  u ,  v ) ,  z >. )
)  =  ( z  e.  ( x  u.  y )  |->  if ( z  e.  x , 
<. z ,  if ( z  =  v ,  w ,  t )
>. ,  <. if ( z  =  w ,  u ,  v ) ,  z >. )
)
18 eqid 2258 . . . 4  |-  if ( z  e.  x , 
<. z ,  if ( z  =  v ,  w ,  t )
>. ,  <. if ( z  =  w ,  u ,  v ) ,  z >. )  =  if ( z  e.  x ,  <. z ,  if ( z  =  v ,  w ,  t ) >. ,  <. if ( z  =  w ,  u ,  v ) ,  z >.
)
1917, 18unxpdomlem3 7037 . . 3  |-  ( ( 1o  ~<  x  /\  1o  ~<  y )  -> 
( x  u.  y
)  ~<_  ( x  X.  y ) )
2010, 16, 19vtocl2g 2822 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( 1o  ~<  A  /\  1o  ~<  B )  ->  ( A  u.  B )  ~<_  ( A  X.  B ) ) )
214, 20mpcom 34 1  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  u.  B
)  ~<_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2763    u. cun 3125   ifcif 3539   <.cop 3617   class class class wbr 3997    e. cmpt 4051    X. cxp 4659   1oc1o 6440    ~<_ cdom 6829    ~< csdm 6830
This theorem is referenced by:  unxpdom2  7039  sucxpdom  7040  cdaxpdom  7783
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 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-1o 6447  df-2o 6448  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834
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