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Theorem unxpdom 7070
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
Dummy variables  x  y  u  t  v  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relsdom 6870 . . . 4  |-  Rel  ~<
21brrelex2i 4730 . . 3  |-  ( 1o 
~<  A  ->  A  e. 
_V )
31brrelex2i 4730 . . 3  |-  ( 1o 
~<  B  ->  B  e. 
_V )
42, 3anim12i 549 . 2  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  e.  _V  /\  B  e.  _V )
)
5 breq2 4027 . . . . 5  |-  ( x  =  A  ->  ( 1o  ~<  x  <->  1o  ~<  A ) )
65anbi1d 685 . . . 4  |-  ( x  =  A  ->  (
( 1o  ~<  x  /\  1o  ~<  y )  <->  ( 1o  ~<  A  /\  1o  ~<  y ) ) )
7 uneq1 3322 . . . . 5  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
8 xpeq1 4703 . . . . 5  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
97, 8breq12d 4036 . . . 4  |-  ( x  =  A  ->  (
( x  u.  y
)  ~<_  ( x  X.  y )  <->  ( A  u.  y )  ~<_  ( A  X.  y ) ) )
106, 9imbi12d 311 . . 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 4027 . . . . 5  |-  ( y  =  B  ->  ( 1o  ~<  y  <->  1o  ~<  B ) )
1211anbi2d 684 . . . 4  |-  ( y  =  B  ->  (
( 1o  ~<  A  /\  1o  ~<  y )  <->  ( 1o  ~<  A  /\  1o  ~<  B ) ) )
13 uneq2 3323 . . . . 5  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
14 xpeq2 4704 . . . . 5  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
1513, 14breq12d 4036 . . . 4  |-  ( y  =  B  ->  (
( A  u.  y
)  ~<_  ( A  X.  y )  <->  ( A  u.  B )  ~<_  ( A  X.  B ) ) )
1612, 15imbi12d 311 . . 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 2283 . . . 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 2283 . . . 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 7069 . . 3  |-  ( ( 1o  ~<  x  /\  1o  ~<  y )  -> 
( x  u.  y
)  ~<_  ( x  X.  y ) )
2010, 16, 19vtocl2g 2847 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( 1o  ~<  A  /\  1o  ~<  B )  ->  ( A  u.  B )  ~<_  ( A  X.  B ) ) )
214, 20mpcom 32 1  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  u.  B
)  ~<_  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150   ifcif 3565   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   1oc1o 6472    ~<_ cdom 6861    ~< csdm 6862
This theorem is referenced by:  unxpdom2  7071  sucxpdom  7072  cdaxpdom  7815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-2o 6480  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866
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