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Theorem cdadom1 7959
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 4318 . . . . 5  |-  { (/) }  e.  _V
21xpdom1 7104 . . . 4  |-  ( A  ~<_  B  ->  ( A  X.  { (/) } )  ~<_  ( B  X.  { (/) } ) )
3 snex 4318 . . . . . 6  |-  { 1o }  e.  _V
4 xpexg 4903 . . . . . 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 7039 . . . . 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 6637 . . . . 5  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
9 undom 7093 . . . . 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 7012 . . . . 5  |-  Rel  ~<_
1312brrelexi 4832 . . . 4  |-  ( A  ~<_  B  ->  A  e.  _V )
14 cdaval 7943 . . . 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 4833 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
17 cdaval 7943 . . . 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 4155 . 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 7942 . . . . . 6  |-  +c  Fn  ( _V  X.  _V )
23 fndm 5448 . . . . . 6  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
2422, 23ax-mp 8 . . . . 5  |-  dom  +c  =  ( _V  X.  _V )
2524ndmov 6131 . . . 4  |-  ( -.  ( A  e.  _V  /\  C  e.  _V )  ->  ( A  +c  C
)  =  (/) )
2621, 25syl 15 . . 3  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  +c  C
)  =  (/) )
27 ovex 6006 . . . 4  |-  ( B  +c  C )  e. 
_V
28270dom 7134 . . 3  |-  (/)  ~<_  ( B  +c  C )
2926, 28syl6eqbr 4162 . 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 1647    e. wcel 1715   _Vcvv 2873    u. cun 3236    i^i cin 3237   (/)c0 3543   {csn 3729   class class class wbr 4125    X. cxp 4790   dom cdm 4792    Fn wfn 5353  (class class class)co 5981   1oc1o 6614    ~<_ cdom 7004    +c ccda 7940
This theorem is referenced by:  cdadom2  7960  cdalepw  7969  unctb  7978  infdif  7982  gchcdaidm  8437  gchhar  8440  gchpwdom  8443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-suc 4501  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-1o 6621  df-en 7007  df-dom 7008  df-cda 7941
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