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Theorem uncdadom 8540
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 4570 . . . . 5  |-  (/)  e.  _V
2 xpsneng 7592 . . . . 5  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  ( A  X.  { (/) } ) 
~~  A )
31, 2mpan2 671 . . . 4  |-  ( A  e.  V  ->  ( A  X.  { (/) } ) 
~~  A )
4 ensym 7554 . . . 4  |-  ( ( A  X.  { (/) } )  ~~  A  ->  A  ~~  ( A  X.  { (/) } ) )
5 endom 7532 . . . 4  |-  ( A 
~~  ( A  X.  { (/) } )  ->  A  ~<_  ( A  X.  { (/) } ) )
63, 4, 53syl 20 . . 3  |-  ( A  e.  V  ->  A  ~<_  ( A  X.  { (/) } ) )
7 1on 7127 . . . . 5  |-  1o  e.  On
8 xpsneng 7592 . . . . 5  |-  ( ( B  e.  W  /\  1o  e.  On )  -> 
( B  X.  { 1o } )  ~~  B
)
97, 8mpan2 671 . . . 4  |-  ( B  e.  W  ->  ( B  X.  { 1o }
)  ~~  B )
10 ensym 7554 . . . 4  |-  ( ( B  X.  { 1o } )  ~~  B  ->  B  ~~  ( B  X.  { 1o }
) )
11 endom 7532 . . . 4  |-  ( B 
~~  ( B  X.  { 1o } )  ->  B  ~<_  ( B  X.  { 1o } ) )
129, 10, 113syl 20 . . 3  |-  ( B  e.  W  ->  B  ~<_  ( B  X.  { 1o } ) )
13 xp01disj 7136 . . . 4  |-  ( ( A  X.  { (/) } )  i^i  ( B  X.  { 1o }
) )  =  (/)
14 undom 7595 . . . 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 671 . . 3  |-  ( ( A  ~<_  ( A  X.  { (/) } )  /\  B  ~<_  ( B  X.  { 1o } ) )  ->  ( A  u.  B )  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
166, 12, 15syl2an 477 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B
)  ~<_  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o } ) ) )
17 cdaval 8539 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +c  B
)  =  ( ( A  X.  { (/) } )  u.  ( B  X.  { 1o }
) ) )
1816, 17breqtrrd 4466 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B
)  ~<_  ( A  +c  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106    u. cun 3467    i^i cin 3468   (/)c0 3778   {csn 4020   class class class wbr 4440   Oncon0 4871    X. cxp 4990  (class class class)co 6275   1oc1o 7113    ~~ cen 7503    ~<_ cdom 7504    +c ccda 8536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508  df-cda 8537
This theorem is referenced by:  cdadom3  8557  unnum  8569  ficardun2  8572  pwsdompw  8573  unctb  8574  infunabs  8576  infcda  8577  infdif  8578
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