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Theorem cdadom1 7696
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 4110 . . . . 5  |-  { (/) }  e.  _V
21xpdom1 6846 . . . 4  |-  ( A  ~<_  B  ->  ( A  X.  { (/) } )  ~<_  ( B  X.  { (/) } ) )
3 snex 4110 . . . . . 6  |-  { 1o }  e.  _V
4 xpexg 4707 . . . . . 6  |-  ( ( C  e.  _V  /\  { 1o }  e.  _V )  ->  ( C  X.  { 1o } )  e. 
_V )
53, 4mpan2 655 . . . . 5  |-  ( C  e.  _V  ->  ( C  X.  { 1o }
)  e.  _V )
6 domrefg 6782 . . . . 5  |-  ( ( C  X.  { 1o } )  e.  _V  ->  ( C  X.  { 1o } )  ~<_  ( C  X.  { 1o }
) )
75, 6syl 17 . . . 4  |-  ( C  e.  _V  ->  ( C  X.  { 1o }
)  ~<_  ( C  X.  { 1o } ) )
8 xp01disj 6381 . . . . 5  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
9 undom 6835 . . . . 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 655 . . . 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 465 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  (
( A  X.  { (/)
} )  u.  ( C  X.  { 1o }
) )  ~<_  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
12 reldom 6755 . . . . 5  |-  Rel  ~<_
1312brrelexi 4636 . . . 4  |-  ( A  ~<_  B  ->  A  e.  _V )
14 cdaval 7680 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A  +c  C
)  =  ( ( A  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
1513, 14sylan 459 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( A  +c  C )  =  ( ( A  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) )
1612brrelex2i 4637 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
17 cdaval 7680 . . . 4  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
1816, 17sylan 459 . . 3  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( B  +c  C )  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) )
1911, 15, 183brtr4d 3950 . 2  |-  ( ( A  ~<_  B  /\  C  e.  _V )  ->  ( A  +c  C )  ~<_  ( B  +c  C ) )
20 simpr 449 . . . . 5  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  -.  C  e.  _V )
2120intnand 887 . . . 4  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  -.  ( A  e. 
_V  /\  C  e.  _V ) )
22 cdafn 7679 . . . . . 6  |-  +c  Fn  ( _V  X.  _V )
23 fndm 5200 . . . . . 6  |-  (  +c  Fn  ( _V  X.  _V )  ->  dom  +c  =  ( _V  X.  _V ) )
2422, 23ax-mp 10 . . . . 5  |-  dom  +c  =  ( _V  X.  _V )
2524ndmov 5856 . . . 4  |-  ( -.  ( A  e.  _V  /\  C  e.  _V )  ->  ( A  +c  C
)  =  (/) )
2621, 25syl 17 . . 3  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  +c  C
)  =  (/) )
27 ovex 5735 . . . 4  |-  ( B  +c  C )  e. 
_V
28270dom 6876 . . 3  |-  (/)  ~<_  ( B  +c  C )
2926, 28syl6eqbr 3957 . 2  |-  ( ( A  ~<_  B  /\  -.  C  e.  _V )  ->  ( A  +c  C
)  ~<_  ( B  +c  C ) )
3019, 29pm2.61dan 769 1  |-  ( A  ~<_  B  ->  ( A  +c  C )  ~<_  ( B  +c  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2727    u. cun 3076    i^i cin 3077   (/)c0 3362   {csn 3544   class class class wbr 3920    X. cxp 4578   dom cdm 4580    Fn wfn 4587  (class class class)co 5710   1oc1o 6358    ~<_ cdom 6747    +c ccda 7677
This theorem is referenced by:  cdadom2  7697  cdalepw  7706  unctb  7715  infdif  7719  gchcdaidm  8170  gchhar  8173  gchpwdom  8176
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-1o 6365  df-en 6750  df-dom 6751  df-cda 7678
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