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Theorem cdadom1 8058
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 4397 . . . . 5  |-  { (/) }  e.  _V
21xpdom1 7199 . . . 4  |-  ( A  ~<_  B  ->  ( A  X.  { (/) } )  ~<_  ( B  X.  { (/) } ) )
3 snex 4397 . . . . . 6  |-  { 1o }  e.  _V
4 xpexg 4981 . . . . . 6  |-  ( ( C  e.  _V  /\  { 1o }  e.  _V )  ->  ( C  X.  { 1o } )  e. 
_V )
53, 4mpan2 653 . . . . 5  |-  ( C  e.  _V  ->  ( C  X.  { 1o }
)  e.  _V )
6 domrefg 7134 . . . . 5  |-  ( ( C  X.  { 1o } )  e.  _V  ->  ( C  X.  { 1o } )  ~<_  ( C  X.  { 1o }
) )
75, 6syl 16 . . . 4  |-  ( C  e.  _V  ->  ( C  X.  { 1o }
)  ~<_  ( C  X.  { 1o } ) )
8 xp01disj 6732 . . . . 5  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
9 undom 7188 . . . . 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 653 . . . 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 464 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  (
( A  X.  { (/)
} )  u.  ( C  X.  { 1o }
) )  ~<_  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
12 reldom 7107 . . . . 5  |-  Rel  ~<_
1312brrelexi 4910 . . . 4  |-  ( A  ~<_  B  ->  A  e.  _V )
14 cdaval 8042 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A  +c  C
)  =  ( ( A  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
1513, 14sylan 458 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( A  +c  C )  =  ( ( A  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) )
1612brrelex2i 4911 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
17 cdaval 8042 . . . 4  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
1816, 17sylan 458 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( B  +c  C )  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) )
1911, 15, 183brtr4d 4234 . 2  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( A  +c  C )  ~<_  ( B  +c  C ) )
20 simpr 448 . . . . 5  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  -.  C  e.  _V )
2120intnand 883 . . . 4  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  -.  ( A  e. 
_V  /\  C  e.  _V ) )
22 cdafn 8041 . . . . . 6  |-  +c  Fn  ( _V  X.  _V )
23 fndm 5536 . . . . . 6  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
2422, 23ax-mp 8 . . . . 5  |-  dom  +c  =  ( _V  X.  _V )
2524ndmov 6223 . . . 4  |-  ( -.  ( A  e.  _V  /\  C  e.  _V )  ->  ( A  +c  C
)  =  (/) )
2621, 25syl 16 . . 3  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  +c  C
)  =  (/) )
27 ovex 6098 . . . 4  |-  ( B  +c  C )  e. 
_V
28270dom 7229 . . 3  |-  (/)  ~<_  ( B  +c  C )
2926, 28syl6eqbr 4241 . 2  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  +c  C
)  ~<_  ( B  +c  C ) )
3019, 29pm2.61dan 767 1  |-  ( A  ~<_  B  ->  ( A  +c  C )  ~<_  ( B  +c  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    u. cun 3310    i^i cin 3311   (/)c0 3620   {csn 3806   class class class wbr 4204    X. cxp 4868   dom cdm 4870    Fn wfn 5441  (class class class)co 6073   1oc1o 6709    ~<_ cdom 7099    +c ccda 8039
This theorem is referenced by:  cdadom2  8059  cdalepw  8068  unctb  8077  infdif  8081  gchcdaidm  8535  gchhar  8538  gchpwdom  8541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-1o 6716  df-en 7102  df-dom 7103  df-cda 8040
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