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Theorem unxpdom 6955
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 6756 . . . 4  |-  Rel  ~<
21brrelex2i 4637 . . 3  |-  ( 1o 
~<  A  ->  A  e. 
_V )
31brrelex2i 4637 . . 3  |-  ( 1o 
~<  B  ->  B  e. 
_V )
42, 3anim12i 551 . 2  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  e.  _V  /\  B  e.  _V )
)
5 breq2 3924 . . . . 5  |-  ( x  =  A  ->  ( 1o  ~<  x  <->  1o  ~<  A ) )
65anbi1d 688 . . . 4  |-  ( x  =  A  ->  (
( 1o  ~<  x  /\  1o  ~<  y )  <->  ( 1o  ~<  A  /\  1o  ~<  y ) ) )
7 uneq1 3232 . . . . 5  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
8 xpeq1 4610 . . . . 5  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
97, 8breq12d 3933 . . . 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 3924 . . . . 5  |-  ( y  =  B  ->  ( 1o  ~<  y  <->  1o  ~<  B ) )
1211anbi2d 687 . . . 4  |-  ( y  =  B  ->  (
( 1o  ~<  A  /\  1o  ~<  y )  <->  ( 1o  ~<  A  /\  1o  ~<  B ) ) )
13 uneq2 3233 . . . . 5  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
14 xpeq2 4611 . . . . 5  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
1513, 14breq12d 3933 . . . 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 2253 . . . 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 2253 . . . 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 6954 . . 3  |-  ( ( 1o  ~<  x  /\  1o  ~<  y )  -> 
( x  u.  y
)  ~<_  ( x  X.  y ) )
2010, 16, 19vtocl2g 2785 . 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 2727    u. cun 3076   ifcif 3470   <.cop 3547   class class class wbr 3920    e. cmpt 3974    X. cxp 4578   1oc1o 6358    ~<_ cdom 6747    ~< csdm 6748
This theorem is referenced by:  unxpdom2  6956  sucxpdom  6957  cdaxpdom  7699
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-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-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-1o 6365  df-2o 6366  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752
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