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Theorem unxpdom2 7087
Description: Corollary of unxpdom 7086. (Contributed by NM, 16-Sep-2004.)
Assertion
Ref Expression
unxpdom2  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  u.  B
)  ~<_  ( A  X.  A ) )

Proof of Theorem unxpdom2
StepHypRef Expression
1 relsdom 6886 . . . . . . . 8  |-  Rel  ~<
21brrelex2i 4746 . . . . . . 7  |-  ( 1o 
~<  A  ->  A  e. 
_V )
32adantr 451 . . . . . 6  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  e.  _V )
4 1onn 6653 . . . . . 6  |-  1o  e.  om
5 xpsneng 6963 . . . . . 6  |-  ( ( A  e.  _V  /\  1o  e.  om )  -> 
( A  X.  { 1o } )  ~~  A
)
63, 4, 5sylancl 643 . . . . 5  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  X.  { 1o } )  ~~  A
)
7 ensym 6926 . . . . 5  |-  ( ( A  X.  { 1o } )  ~~  A  ->  A  ~~  ( A  X.  { 1o }
) )
86, 7syl 15 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~~  ( A  X.  { 1o } ) )
9 endom 6904 . . . 4  |-  ( A 
~~  ( A  X.  { 1o } )  ->  A  ~<_  ( A  X.  { 1o } ) )
108, 9syl 15 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~<_  ( A  X.  { 1o } ) )
11 simpr 447 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  B  ~<_  A )
12 0ex 4166 . . . . . 6  |-  (/)  e.  _V
13 xpsneng 6963 . . . . . 6  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
143, 12, 13sylancl 643 . . . . 5  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  X.  { (/)
} )  ~~  A
)
15 ensym 6926 . . . . 5  |-  ( ( A  X.  { (/) } )  ~~  A  ->  A  ~~  ( A  X.  { (/) } ) )
1614, 15syl 15 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~~  ( A  X.  { (/) } ) )
17 domentr 6936 . . . 4  |-  ( ( B  ~<_  A  /\  A  ~~  ( A  X.  { (/)
} ) )  ->  B  ~<_  ( A  X.  { (/) } ) )
1811, 16, 17syl2anc 642 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  B  ~<_  ( A  X.  { (/) } ) )
19 1n0 6510 . . . 4  |-  1o  =/=  (/)
20 xpsndisj 5119 . . . 4  |-  ( 1o  =/=  (/)  ->  ( ( A  X.  { 1o }
)  i^i  ( A  X.  { (/) } ) )  =  (/) )
2119, 20mp1i 11 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( ( A  X.  { 1o } )  i^i  ( A  X.  { (/)
} ) )  =  (/) )
22 undom 6966 . . 3  |-  ( ( ( A  ~<_  ( A  X.  { 1o }
)  /\  B  ~<_  ( A  X.  { (/) } ) )  /\  ( ( A  X.  { 1o } )  i^i  ( A  X.  { (/) } ) )  =  (/) )  -> 
( A  u.  B
)  ~<_  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) ) )
2310, 18, 21, 22syl21anc 1181 . 2  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  u.  B
)  ~<_  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) ) )
24 sdomentr 7011 . . . . 5  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { 1o } ) )  ->  1o  ~<  ( A  X.  { 1o }
) )
258, 24syldan 456 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  1o  ~<  ( A  X.  { 1o } ) )
26 sdomentr 7011 . . . . 5  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  1o  ~<  ( A  X.  { (/) } ) )
2716, 26syldan 456 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  1o  ~<  ( A  X.  { (/) } ) )
28 unxpdom 7086 . . . 4  |-  ( ( 1o  ~<  ( A  X.  { 1o } )  /\  1o  ~<  ( A  X.  { (/) } ) )  ->  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) )  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )
2925, 27, 28syl2anc 642 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( ( A  X.  { 1o } )  u.  ( A  X.  { (/)
} ) )  ~<_  ( ( A  X.  { 1o } )  X.  ( A  X.  { (/) } ) ) )
30 xpen 7040 . . . 4  |-  ( ( ( A  X.  { 1o } )  ~~  A  /\  ( A  X.  { (/)
} )  ~~  A
)  ->  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) )  ~~  ( A  X.  A ) )
316, 14, 30syl2anc 642 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( ( A  X.  { 1o } )  X.  ( A  X.  { (/)
} ) )  ~~  ( A  X.  A
) )
32 domentr 6936 . . 3  |-  ( ( ( ( A  X.  { 1o } )  u.  ( A  X.  { (/)
} ) )  ~<_  ( ( A  X.  { 1o } )  X.  ( A  X.  { (/) } ) )  /\  ( ( A  X.  { 1o } )  X.  ( A  X.  { (/) } ) )  ~~  ( A  X.  A ) )  ->  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) )  ~<_  ( A  X.  A ) )
3329, 31, 32syl2anc 642 . 2  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( ( A  X.  { 1o } )  u.  ( A  X.  { (/)
} ) )  ~<_  ( A  X.  A ) )
34 domtr 6930 . 2  |-  ( ( ( A  u.  B
)  ~<_  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) )  /\  ( ( A  X.  { 1o } )  u.  ( A  X.  { (/) } ) )  ~<_  ( A  X.  A ) )  -> 
( A  u.  B
)  ~<_  ( A  X.  A ) )
3523, 33, 34syl2anc 642 1  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  u.  B
)  ~<_  ( A  X.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   class class class wbr 4039   omcom 4672    X. cxp 4703   1oc1o 6488    ~~ cen 6876    ~<_ cdom 6877    ~< csdm 6878
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-csb 3095  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-int 3879  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-1st 6138  df-2nd 6139  df-1o 6495  df-2o 6496  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882
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