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Theorem unxpdom2 7725
Description: Corollary of unxpdom 7724. (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 7520 . . . . . . . 8  |-  Rel  ~<
21brrelex2i 5040 . . . . . . 7  |-  ( 1o 
~<  A  ->  A  e. 
_V )
32adantr 465 . . . . . 6  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  e.  _V )
4 1onn 7285 . . . . . 6  |-  1o  e.  om
5 xpsneng 7599 . . . . . 6  |-  ( ( A  e.  _V  /\  1o  e.  om )  -> 
( A  X.  { 1o } )  ~~  A
)
63, 4, 5sylancl 662 . . . . 5  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  X.  { 1o } )  ~~  A
)
76ensymd 7563 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~~  ( A  X.  { 1o } ) )
8 endom 7539 . . . 4  |-  ( A 
~~  ( A  X.  { 1o } )  ->  A  ~<_  ( A  X.  { 1o } ) )
97, 8syl 16 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~<_  ( A  X.  { 1o } ) )
10 simpr 461 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  B  ~<_  A )
11 0ex 4577 . . . . . 6  |-  (/)  e.  _V
12 xpsneng 7599 . . . . . 6  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
133, 11, 12sylancl 662 . . . . 5  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  X.  { (/)
} )  ~~  A
)
1413ensymd 7563 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~~  ( A  X.  { (/) } ) )
15 domentr 7571 . . . 4  |-  ( ( B  ~<_  A  /\  A  ~~  ( A  X.  { (/)
} ) )  ->  B  ~<_  ( A  X.  { (/) } ) )
1610, 14, 15syl2anc 661 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  B  ~<_  ( A  X.  { (/) } ) )
17 1n0 7142 . . . 4  |-  1o  =/=  (/)
18 xpsndisj 5428 . . . 4  |-  ( 1o  =/=  (/)  ->  ( ( A  X.  { 1o }
)  i^i  ( A  X.  { (/) } ) )  =  (/) )
1917, 18mp1i 12 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( ( A  X.  { 1o } )  i^i  ( A  X.  { (/)
} ) )  =  (/) )
20 undom 7602 . . 3  |-  ( ( ( A  ~<_  ( A  X.  { 1o }
)  /\  B  ~<_  ( A  X.  { (/) } ) )  /\  ( ( A  X.  { 1o } )  i^i  ( A  X.  { (/) } ) )  =  (/) )  -> 
( A  u.  B
)  ~<_  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) ) )
219, 16, 19, 20syl21anc 1227 . 2  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  u.  B
)  ~<_  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) ) )
22 sdomentr 7648 . . . . 5  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { 1o } ) )  ->  1o  ~<  ( A  X.  { 1o }
) )
237, 22syldan 470 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  1o  ~<  ( A  X.  { 1o } ) )
24 sdomentr 7648 . . . . 5  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  1o  ~<  ( A  X.  { (/) } ) )
2514, 24syldan 470 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  1o  ~<  ( A  X.  { (/) } ) )
26 unxpdom 7724 . . . 4  |-  ( ( 1o  ~<  ( A  X.  { 1o } )  /\  1o  ~<  ( A  X.  { (/) } ) )  ->  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) )  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )
2723, 25, 26syl2anc 661 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( ( A  X.  { 1o } )  u.  ( A  X.  { (/)
} ) )  ~<_  ( ( A  X.  { 1o } )  X.  ( A  X.  { (/) } ) ) )
28 xpen 7677 . . . 4  |-  ( ( ( A  X.  { 1o } )  ~~  A  /\  ( A  X.  { (/)
} )  ~~  A
)  ->  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) )  ~~  ( A  X.  A ) )
296, 13, 28syl2anc 661 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( ( A  X.  { 1o } )  X.  ( A  X.  { (/)
} ) )  ~~  ( A  X.  A
) )
30 domentr 7571 . . 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 ) )
3127, 29, 30syl2anc 661 . 2  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( ( A  X.  { 1o } )  u.  ( A  X.  { (/)
} ) )  ~<_  ( A  X.  A ) )
32 domtr 7565 . 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 ) )
3321, 31, 32syl2anc 661 1  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  u.  B
)  ~<_  ( A  X.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    u. cun 3474    i^i cin 3475   (/)c0 3785   {csn 4027   class class class wbr 4447    X. cxp 4997   omcom 6678   1oc1o 7120    ~~ cen 7510    ~<_ cdom 7511    ~< csdm 7512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-1st 6781  df-2nd 6782  df-1o 7127  df-2o 7128  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516
This theorem is referenced by: (None)
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