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Theorem unxpdom 7086
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 6886 . . . 4  |-  Rel  ~<
21brrelex2i 4746 . . 3  |-  ( 1o 
~<  A  ->  A  e. 
_V )
31brrelex2i 4746 . . 3  |-  ( 1o 
~<  B  ->  B  e. 
_V )
42, 3anim12i 549 . 2  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  e.  _V  /\  B  e.  _V )
)
5 breq2 4043 . . . . 5  |-  ( x  =  A  ->  ( 1o  ~<  x  <->  1o  ~<  A ) )
65anbi1d 685 . . . 4  |-  ( x  =  A  ->  (
( 1o  ~<  x  /\  1o  ~<  y )  <->  ( 1o  ~<  A  /\  1o  ~<  y ) ) )
7 uneq1 3335 . . . . 5  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
8 xpeq1 4719 . . . . 5  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
97, 8breq12d 4052 . . . 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 4043 . . . . 5  |-  ( y  =  B  ->  ( 1o  ~<  y  <->  1o  ~<  B ) )
1211anbi2d 684 . . . 4  |-  ( y  =  B  ->  (
( 1o  ~<  A  /\  1o  ~<  y )  <->  ( 1o  ~<  A  /\  1o  ~<  B ) ) )
13 uneq2 3336 . . . . 5  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
14 xpeq2 4720 . . . . 5  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
1513, 14breq12d 4052 . . . 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 2296 . . . 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 2296 . . . 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 7085 . . 3  |-  ( ( 1o  ~<  x  /\  1o  ~<  y )  -> 
( x  u.  y
)  ~<_  ( x  X.  y ) )
2010, 16, 19vtocl2g 2860 . 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 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163   ifcif 3578   <.cop 3656   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   1oc1o 6488    ~<_ cdom 6877    ~< csdm 6878
This theorem is referenced by:  unxpdom2  7087  sucxpdom  7088  cdaxpdom  7831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-2o 6496  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882
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