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Theorem cdadom1 7808
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 4215 . . . . 5  |-  { (/) }  e.  _V
21xpdom1 6957 . . . 4  |-  ( A  ~<_  B  ->  ( A  X.  { (/) } )  ~<_  ( B  X.  { (/) } ) )
3 snex 4215 . . . . . 6  |-  { 1o }  e.  _V
4 xpexg 4799 . . . . . 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 6892 . . . . 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 6491 . . . . 5  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
9 undom 6946 . . . . 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 6865 . . . . 5  |-  Rel  ~<_
1312brrelexi 4728 . . . 4  |-  ( A  ~<_  B  ->  A  e.  _V )
14 cdaval 7792 . . . 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 4729 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
17 cdaval 7792 . . . 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 4054 . 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 7791 . . . . . 6  |-  +c  Fn  ( _V  X.  _V )
23 fndm 5309 . . . . . 6  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
2422, 23ax-mp 8 . . . . 5  |-  dom  +c  =  ( _V  X.  _V )
2524ndmov 5966 . . . 4  |-  ( -.  ( A  e.  _V  /\  C  e.  _V )  ->  ( A  +c  C
)  =  (/) )
2621, 25syl 15 . . 3  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  +c  C
)  =  (/) )
27 ovex 5845 . . . 4  |-  ( B  +c  C )  e. 
_V
28270dom 6987 . . 3  |-  (/)  ~<_  ( B  +c  C )
2926, 28syl6eqbr 4061 . 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 1623    e. wcel 1685   _Vcvv 2789    u. cun 3151    i^i cin 3152   (/)c0 3456   {csn 3641   class class class wbr 4024    X. cxp 4686    dom cdm 4688    Fn wfn 5216  (class class class)co 5820   1oc1o 6468    ~<_ cdom 6857    +c ccda 7789
This theorem is referenced by:  cdadom2  7809  cdalepw  7818  unctb  7827  infdif  7831  gchcdaidm  8286  gchhar  8289  gchpwdom  8292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-1o 6475  df-en 6860  df-dom 6861  df-cda 7790
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