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Theorem unxpdom2 7319
Description: Corollary of unxpdom 7318. (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 7118 . . . . . . . 8  |-  Rel  ~<
21brrelex2i 4921 . . . . . . 7  |-  ( 1o 
~<  A  ->  A  e. 
_V )
32adantr 453 . . . . . 6  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  e.  _V )
4 1onn 6884 . . . . . 6  |-  1o  e.  om
5 xpsneng 7195 . . . . . 6  |-  ( ( A  e.  _V  /\  1o  e.  om )  -> 
( A  X.  { 1o } )  ~~  A
)
63, 4, 5sylancl 645 . . . . 5  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  X.  { 1o } )  ~~  A
)
76ensymd 7160 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~~  ( A  X.  { 1o } ) )
8 endom 7136 . . . 4  |-  ( A 
~~  ( A  X.  { 1o } )  ->  A  ~<_  ( A  X.  { 1o } ) )
97, 8syl 16 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~<_  ( A  X.  { 1o } ) )
10 simpr 449 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  B  ~<_  A )
11 0ex 4341 . . . . . 6  |-  (/)  e.  _V
12 xpsneng 7195 . . . . . 6  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
133, 11, 12sylancl 645 . . . . 5  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  X.  { (/)
} )  ~~  A
)
1413ensymd 7160 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~~  ( A  X.  { (/) } ) )
15 domentr 7168 . . . 4  |-  ( ( B  ~<_  A  /\  A  ~~  ( A  X.  { (/)
} ) )  ->  B  ~<_  ( A  X.  { (/) } ) )
1610, 14, 15syl2anc 644 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  B  ~<_  ( A  X.  { (/) } ) )
17 1n0 6741 . . . 4  |-  1o  =/=  (/)
18 xpsndisj 5298 . . . 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 7198 . . 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 1184 . 2  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  u.  B
)  ~<_  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) ) )
22 sdomentr 7243 . . . . 5  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { 1o } ) )  ->  1o  ~<  ( A  X.  { 1o }
) )
237, 22syldan 458 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  1o  ~<  ( A  X.  { 1o } ) )
24 sdomentr 7243 . . . . 5  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  1o  ~<  ( A  X.  { (/) } ) )
2514, 24syldan 458 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  1o  ~<  ( A  X.  { (/) } ) )
26 unxpdom 7318 . . . 4  |-  ( ( 1o  ~<  ( A  X.  { 1o } )  /\  1o  ~<  ( A  X.  { (/) } ) )  ->  ( ( A  X.  { 1o }
)  u.  ( A  X.  { (/) } ) )  ~<_  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) ) )
2723, 25, 26syl2anc 644 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( ( A  X.  { 1o } )  u.  ( A  X.  { (/)
} ) )  ~<_  ( ( A  X.  { 1o } )  X.  ( A  X.  { (/) } ) ) )
28 xpen 7272 . . . 4  |-  ( ( ( A  X.  { 1o } )  ~~  A  /\  ( A  X.  { (/)
} )  ~~  A
)  ->  ( ( A  X.  { 1o }
)  X.  ( A  X.  { (/) } ) )  ~~  ( A  X.  A ) )
296, 13, 28syl2anc 644 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( ( A  X.  { 1o } )  X.  ( A  X.  { (/)
} ) )  ~~  ( A  X.  A
) )
30 domentr 7168 . . 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 644 . 2  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( ( A  X.  { 1o } )  u.  ( A  X.  { (/)
} ) )  ~<_  ( A  X.  A ) )
32 domtr 7162 . 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 644 1  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  u.  B
)  ~<_  ( A  X.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958    u. cun 3320    i^i cin 3321   (/)c0 3630   {csn 3816   class class class wbr 4214   omcom 4847    X. cxp 4878   1oc1o 6719    ~~ cen 7108    ~<_ cdom 7109    ~< csdm 7110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-1st 6351  df-2nd 6352  df-1o 6726  df-2o 6727  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114
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