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Theorem unxpdom2 7747
Description: Corollary of unxpdom 7746. (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 7542 . . . . . . . 8  |-  Rel  ~<
21brrelex2i 5050 . . . . . . 7  |-  ( 1o 
~<  A  ->  A  e. 
_V )
32adantr 465 . . . . . 6  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  e.  _V )
4 1onn 7306 . . . . . 6  |-  1o  e.  om
5 xpsneng 7621 . . . . . 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 7585 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~~  ( A  X.  { 1o } ) )
8 endom 7561 . . . 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 4587 . . . . . 6  |-  (/)  e.  _V
12 xpsneng 7621 . . . . . 6  |-  ( ( A  e.  _V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
133, 11, 12sylancl 662 . . . . 5  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  -> 
( A  X.  { (/)
} )  ~~  A
)
1413ensymd 7585 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  A  ~~  ( A  X.  { (/) } ) )
15 domentr 7593 . . . 4  |-  ( ( B  ~<_  A  /\  A  ~~  ( A  X.  { (/)
} ) )  ->  B  ~<_  ( A  X.  { (/) } ) )
1610, 14, 15syl2anc 661 . . 3  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  B  ~<_  ( A  X.  { (/) } ) )
17 1n0 7163 . . . 4  |-  1o  =/=  (/)
18 xpsndisj 5437 . . . 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 7624 . . 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 7670 . . . . 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 7670 . . . . 5  |-  ( ( 1o  ~<  A  /\  A  ~~  ( A  X.  { (/) } ) )  ->  1o  ~<  ( A  X.  { (/) } ) )
2514, 24syldan 470 . . . 4  |-  ( ( 1o  ~<  A  /\  B  ~<_  A )  ->  1o  ~<  ( A  X.  { (/) } ) )
26 unxpdom 7746 . . . 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 7699 . . . 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 7593 . . 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 7587 . 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 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    u. cun 3469    i^i cin 3470   (/)c0 3793   {csn 4032   class class class wbr 4456    X. cxp 5006   omcom 6699   1oc1o 7141    ~~ cen 7532    ~<_ cdom 7533    ~< csdm 7534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-1st 6799  df-2nd 6800  df-1o 7148  df-2o 7149  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538
This theorem is referenced by: (None)
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