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Theorem unxpdom2 7071
Description: Corollary of unxpdom 7070. (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 6870 . . . . . . . 8  |-  Rel  ~<
21brrelex2i 4730 . . . . . . 7  |-  ( 1o 
~<  A  ->  A  e. 
_V )
32adantr 451 . . . . . 6  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  e.  _V )
4 1onn 6637 . . . . . 6  |-  1o  e.  om
5 xpsneng 6947 . . . . . 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 6910 . . . . 5  |-  ( ( A  X.  { 1o } )  ~~  A  ->  A  ~~  ( A  X.  { 1o }
) )
86, 7syl 15 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~~  ( A  X.  { 1o } ) )
9 endom 6888 . . . 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 4150 . . . . . 6  |-  (/)  e.  _V
13 xpsneng 6947 . . . . . 6  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
143, 12, 13sylancl 643 . . . . 5  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  X.  { (/)
} )  ~~  A
)
15 ensym 6910 . . . . 5  |-  ( ( A  X.  { (/) } )  ~~  A  ->  A  ~~  ( A  X.  { (/) } ) )
1614, 15syl 15 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~~  ( A  X.  { (/) } ) )
17 domentr 6920 . . . 4  |-  ( ( B  ~<_  A  /\  A  ~~  ( A  X.  { (/)
} ) )  ->  B  ~<_  ( A  X.  { (/) } ) )
1811, 16, 17syl2anc 642 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  B  ~<_  ( A  X.  { (/) } ) )
19 1n0 6494 . . . 4  |-  1o  =/=  (/)
20 xpsndisj 5103 . . . 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 6950 . . 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 6995 . . . . 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 6995 . . . . 5  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  1o  ~<  ( A  X.  { (/) } ) )
2716, 26syldan 456 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  1o  ~<  ( A  X.  { (/) } ) )
28 unxpdom 7070 . . . 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 7024 . . . 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 6920 . . 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 6914 . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   class class class wbr 4023   omcom 4656    X. cxp 4687   1oc1o 6472    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6122  df-2nd 6123  df-1o 6479  df-2o 6480  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866
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