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Theorem infcdaabs 8042
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 8023 . . . . . 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 8016 . . . . . 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 4198 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  +c  B )  ~<_  ( A  X.  2o ) )
7 2onn 6842 . . . . . . 7  |-  2o  e.  om
8 nnsdom 7564 . . . . . . 7  |-  ( 2o  e.  om  ->  2o  ~<  om )
9 sdomdom 7094 . . . . . . 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 7119 . . . . . 6  |-  ( ( 2o  ~<_  om  /\  om  ~<_  A )  ->  2o  ~<_  A )
1310, 11, 12sylancr 645 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  2o  ~<_  A )
14 xpdom2g 7163 . . . . 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 7119 . . . 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 7854 . . . 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 7125 . . 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 7074 . . . . 5  |-  Rel  ~<_
2322brrelexi 4877 . . . 4  |-  ( B  ~<_  A  ->  B  e.  _V )
24233ad2ant3 980 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  B  e.  _V )
25 cdadom3 8024 . . 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 7186 . 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 1649    e. wcel 1721   _Vcvv 2916   class class class wbr 4172   omcom 4804    X. cxp 4835   dom cdm 4837  (class class class)co 6040   2oc2o 6677    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067   cardccrd 7778    +c ccda 8003
This theorem is referenced by:  infunabs  8043  infcda  8044  infdif  8045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-cda 8004
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