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Theorem unxpdom 7785
Description: Cartesian 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 7584 . . . 4  |-  Rel  ~<
21brrelex2i 4896 . . 3  |-  ( 1o 
~<  A  ->  A  e. 
_V )
31brrelex2i 4896 . . 3  |-  ( 1o 
~<  B  ->  B  e. 
_V )
42, 3anim12i 568 . 2  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  e.  _V  /\  B  e.  _V )
)
5 breq2 4430 . . . . 5  |-  ( x  =  A  ->  ( 1o  ~<  x  <->  1o  ~<  A ) )
65anbi1d 709 . . . 4  |-  ( x  =  A  ->  (
( 1o  ~<  x  /\  1o  ~<  y )  <->  ( 1o  ~<  A  /\  1o  ~<  y ) ) )
7 uneq1 3619 . . . . 5  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
8 xpeq1 4868 . . . . 5  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
97, 8breq12d 4439 . . . 4  |-  ( x  =  A  ->  (
( x  u.  y
)  ~<_  ( x  X.  y )  <->  ( A  u.  y )  ~<_  ( A  X.  y ) ) )
106, 9imbi12d 321 . . 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 4430 . . . . 5  |-  ( y  =  B  ->  ( 1o  ~<  y  <->  1o  ~<  B ) )
1211anbi2d 708 . . . 4  |-  ( y  =  B  ->  (
( 1o  ~<  A  /\  1o  ~<  y )  <->  ( 1o  ~<  A  /\  1o  ~<  B ) ) )
13 uneq2 3620 . . . . 5  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
14 xpeq2 4869 . . . . 5  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
1513, 14breq12d 4439 . . . 4  |-  ( y  =  B  ->  (
( A  u.  y
)  ~<_  ( A  X.  y )  <->  ( A  u.  B )  ~<_  ( A  X.  B ) ) )
1612, 15imbi12d 321 . . 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 2429 . . . 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 2429 . . . 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 7784 . . 3  |-  ( ( 1o  ~<  x  /\  1o  ~<  y )  -> 
( x  u.  y
)  ~<_  ( x  X.  y ) )
2010, 16, 19vtocl2g 3149 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( 1o  ~<  A  /\  1o  ~<  B )  ->  ( A  u.  B )  ~<_  ( A  X.  B ) ) )
214, 20mpcom 37 1  |-  ( ( 1o  ~<  A  /\  1o  ~<  B )  -> 
( A  u.  B
)  ~<_  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    u. cun 3440   ifcif 3915   <.cop 4008   class class class wbr 4426    |-> cmpt 4484    X. cxp 4852   1oc1o 7183    ~<_ cdom 7575    ~< csdm 7576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-1o 7190  df-2o 7191  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580
This theorem is referenced by:  unxpdom2  7786  sucxpdom  7787  cdaxpdom  8617
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