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Theorem uncdadom 7985
Description: Cardinal addition dominates union. (Contributed by NM, 28-Sep-2004.)
Assertion
Ref Expression
uncdadom  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B
)  ~<_  ( A  +c  B ) )

Proof of Theorem uncdadom
StepHypRef Expression
1 0ex 4281 . . . . 5  |-  (/)  e.  _V
2 xpsneng 7130 . . . . 5  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
31, 2mpan2 653 . . . 4  |-  ( A  e.  V  ->  ( A  X.  { (/) } ) 
~~  A )
4 ensym 7093 . . . 4  |-  ( ( A  X.  { (/) } )  ~~  A  ->  A  ~~  ( A  X.  { (/) } ) )
5 endom 7071 . . . 4  |-  ( A 
~~  ( A  X.  { (/) } )  ->  A  ~<_  ( A  X.  { (/) } ) )
63, 4, 53syl 19 . . 3  |-  ( A  e.  V  ->  A  ~<_  ( A  X.  { (/) } ) )
7 1on 6668 . . . . 5  |-  1o  e.  On
8 xpsneng 7130 . . . . 5  |-  ( ( B  e.  W  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
97, 8mpan2 653 . . . 4  |-  ( B  e.  W  ->  ( B  X.  { 1o }
)  ~~  B )
10 ensym 7093 . . . 4  |-  ( ( B  X.  { 1o } )  ~~  B  ->  B  ~~  ( B  X.  { 1o }
) )
11 endom 7071 . . . 4  |-  ( B 
~~  ( B  X.  { 1o } )  ->  B  ~<_  ( B  X.  { 1o } ) )
129, 10, 113syl 19 . . 3  |-  ( B  e.  W  ->  B  ~<_  ( B  X.  { 1o } ) )
13 xp01disj 6677 . . . 4  |-  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) )  =  (/)
14 undom 7133 . . . 4  |-  ( ( ( A  ~<_  ( A  X.  { (/) } )  /\  B  ~<_  ( B  X.  { 1o }
) )  /\  (
( A  X.  { (/)
} )  i^i  ( B  X.  { 1o }
) )  =  (/) )  ->  ( A  u.  B )  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
1513, 14mpan2 653 . . 3  |-  ( ( A  ~<_  ( A  X.  { (/) } )  /\  B  ~<_  ( B  X.  { 1o } ) )  ->  ( A  u.  B )  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
166, 12, 15syl2an 464 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B
)  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
17 cdaval 7984 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
1816, 17breqtrrd 4180 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B
)  ~<_  ( A  +c  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900    u. cun 3262    i^i cin 3263   (/)c0 3572   {csn 3758   class class class wbr 4154   Oncon0 4523    X. cxp 4817  (class class class)co 6021   1oc1o 6654    ~~ cen 7043    ~<_ cdom 7044    +c ccda 7981
This theorem is referenced by:  cdadom3  8002  unnum  8014  ficardun2  8017  pwsdompw  8018  unctb  8019  infunabs  8021  infcda  8022  infdif  8023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-suc 4529  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1o 6661  df-er 6842  df-en 7047  df-dom 7048  df-cda 7982
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