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Theorem cdadom1 7828
Description: Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cdadom1  |-  ( A  ~<_  B  ->  ( A  +c  C )  ~<_  ( B  +c  C ) )

Proof of Theorem cdadom1
StepHypRef Expression
1 snex 4232 . . . . 5  |-  { (/) }  e.  _V
21xpdom1 6977 . . . 4  |-  ( A  ~<_  B  ->  ( A  X.  { (/) } )  ~<_  ( B  X.  { (/) } ) )
3 snex 4232 . . . . . 6  |-  { 1o }  e.  _V
4 xpexg 4816 . . . . . 6  |-  ( ( C  e.  _V  /\  { 1o }  e.  _V )  ->  ( C  X.  { 1o } )  e. 
_V )
53, 4mpan2 652 . . . . 5  |-  ( C  e.  _V  ->  ( C  X.  { 1o }
)  e.  _V )
6 domrefg 6912 . . . . 5  |-  ( ( C  X.  { 1o } )  e.  _V  ->  ( C  X.  { 1o } )  ~<_  ( C  X.  { 1o }
) )
75, 6syl 15 . . . 4  |-  ( C  e.  _V  ->  ( C  X.  { 1o }
)  ~<_  ( C  X.  { 1o } ) )
8 xp01disj 6511 . . . . 5  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
9 undom 6966 . . . . 5  |-  ( ( ( ( A  X.  { (/) } )  ~<_  ( B  X.  { (/) } )  /\  ( C  X.  { 1o }
)  ~<_  ( C  X.  { 1o } ) )  /\  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) )  =  (/) )  ->  (
( A  X.  { (/)
} )  u.  ( C  X.  { 1o }
) )  ~<_  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
108, 9mpan2 652 . . . 4  |-  ( ( ( A  X.  { (/)
} )  ~<_  ( B  X.  { (/) } )  /\  ( C  X.  { 1o } )  ~<_  ( C  X.  { 1o } ) )  -> 
( ( A  X.  { (/) } )  u.  ( C  X.  { 1o } ) )  ~<_  ( ( B  X.  { (/)
} )  u.  ( C  X.  { 1o }
) ) )
112, 7, 10syl2an 463 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  (
( A  X.  { (/)
} )  u.  ( C  X.  { 1o }
) )  ~<_  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
12 reldom 6885 . . . . 5  |-  Rel  ~<_
1312brrelexi 4745 . . . 4  |-  ( A  ~<_  B  ->  A  e.  _V )
14 cdaval 7812 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A  +c  C
)  =  ( ( A  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
1513, 14sylan 457 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( A  +c  C )  =  ( ( A  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) )
1612brrelex2i 4746 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
17 cdaval 7812 . . . 4  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
1816, 17sylan 457 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( B  +c  C )  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) )
1911, 15, 183brtr4d 4069 . 2  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( A  +c  C )  ~<_  ( B  +c  C ) )
20 simpr 447 . . . . 5  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  -.  C  e.  _V )
2120intnand 882 . . . 4  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  -.  ( A  e. 
_V  /\  C  e.  _V ) )
22 cdafn 7811 . . . . . 6  |-  +c  Fn  ( _V  X.  _V )
23 fndm 5359 . . . . . 6  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
2422, 23ax-mp 8 . . . . 5  |-  dom  +c  =  ( _V  X.  _V )
2524ndmov 6020 . . . 4  |-  ( -.  ( A  e.  _V  /\  C  e.  _V )  ->  ( A  +c  C
)  =  (/) )
2621, 25syl 15 . . 3  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  +c  C
)  =  (/) )
27 ovex 5899 . . . 4  |-  ( B  +c  C )  e. 
_V
28270dom 7007 . . 3  |-  (/)  ~<_  ( B  +c  C )
2926, 28syl6eqbr 4076 . 2  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  +c  C
)  ~<_  ( B  +c  C ) )
3019, 29pm2.61dan 766 1  |-  ( A  ~<_  B  ->  ( A  +c  C )  ~<_  ( B  +c  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   class class class wbr 4039    X. cxp 4703   dom cdm 4705    Fn wfn 5266  (class class class)co 5874   1oc1o 6488    ~<_ cdom 6877    +c ccda 7809
This theorem is referenced by:  cdadom2  7829  cdalepw  7838  unctb  7847  infdif  7851  gchcdaidm  8306  gchhar  8309  gchpwdom  8312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-1o 6495  df-en 6880  df-dom 6881  df-cda 7810
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