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Theorem infcdaabs 8078
Description: Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infcdaabs  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  +c  B )  ~~  A )

Proof of Theorem infcdaabs
StepHypRef Expression
1 cdadom2 8059 . . . . . 6  |-  ( B  ~<_  A  ->  ( A  +c  B )  ~<_  ( A  +c  A ) )
213ad2ant3 980 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  +c  B )  ~<_  ( A  +c  A ) )
3 simp1 957 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  A  e.  dom  card )
4 xp2cda 8052 . . . . . 6  |-  ( A  e.  dom  card  ->  ( A  X.  2o )  =  ( A  +c  A ) )
53, 4syl 16 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  X.  2o )  =  ( A  +c  A
) )
62, 5breqtrrd 4230 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  +c  B )  ~<_  ( A  X.  2o ) )
7 2onn 6875 . . . . . . 7  |-  2o  e.  om
8 nnsdom 7600 . . . . . . 7  |-  ( 2o  e.  om  ->  2o  ~<  om )
9 sdomdom 7127 . . . . . . 7  |-  ( 2o 
~<  om  ->  2o  ~<_  om )
107, 8, 9mp2b 10 . . . . . 6  |-  2o  ~<_  om
11 simp2 958 . . . . . 6  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  om  ~<_  A )
12 domtr 7152 . . . . . 6  |-  ( ( 2o  ~<_  om  /\  om  ~<_  A )  ->  2o  ~<_  A )
1310, 11, 12sylancr 645 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  2o  ~<_  A )
14 xpdom2g 7196 . . . . 5  |-  ( ( A  e.  dom  card  /\  2o  ~<_  A )  -> 
( A  X.  2o )  ~<_  ( A  X.  A ) )
153, 13, 14syl2anc 643 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  X.  2o )  ~<_  ( A  X.  A ) )
16 domtr 7152 . . . 4  |-  ( ( ( A  +c  B
)  ~<_  ( A  X.  2o )  /\  ( A  X.  2o )  ~<_  ( A  X.  A ) )  ->  ( A  +c  B )  ~<_  ( A  X.  A ) )
176, 15, 16syl2anc 643 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  +c  B )  ~<_  ( A  X.  A ) )
18 infxpidm2 7890 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  A )
19183adant3 977 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  X.  A )  ~~  A )
20 domentr 7158 . . 3  |-  ( ( ( A  +c  B
)  ~<_  ( A  X.  A )  /\  ( A  X.  A )  ~~  A )  ->  ( A  +c  B )  ~<_  A )
2117, 19, 20syl2anc 643 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  +c  B )  ~<_  A )
22 reldom 7107 . . . . 5  |-  Rel  ~<_
2322brrelexi 4910 . . . 4  |-  ( B  ~<_  A  ->  B  e.  _V )
24233ad2ant3 980 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  B  e.  _V )
25 cdadom3 8060 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  _V )  ->  A  ~<_  ( A  +c  B ) )
263, 24, 25syl2anc 643 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  A  ~<_  ( A  +c  B
) )
27 sbth 7219 . 2  |-  ( ( ( A  +c  B
)  ~<_  A  /\  A  ~<_  ( A  +c  B
) )  ->  ( A  +c  B )  ~~  A )
2821, 26, 27syl2anc 643 1  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  +c  B )  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2948   class class class wbr 4204   omcom 4837    X. cxp 4868   dom cdm 4870  (class class class)co 6073   2oc2o 6710    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100   cardccrd 7814    +c ccda 8039
This theorem is referenced by:  infunabs  8079  infcda  8080  infdif  8081
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-reu 2704  df-rmo 2705  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-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  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-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-cda 8040
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