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Theorem infcda 7788
Description: The sum of two cardinal numbers is their maximum, if one of them is infinite. Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infcda  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  +c  B
)  ~~  ( A  u.  B ) )

Proof of Theorem infcda
StepHypRef Expression
1 unnum 7780 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  u.  B )  e.  dom  card )
213adant3 980 . . . . . 6  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  u.  B
)  e.  dom  card )
3 ssun2 3300 . . . . . 6  |-  B  C_  ( A  u.  B
)
4 ssdomg 6861 . . . . . 6  |-  ( ( A  u.  B )  e.  dom  card  ->  ( B  C_  ( A  u.  B )  ->  B  ~<_  ( A  u.  B
) ) )
52, 3, 4ee10 1372 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  B  ~<_  ( A  u.  B ) )
6 cdadom2 7767 . . . . 5  |-  ( B  ~<_  ( A  u.  B
)  ->  ( A  +c  B )  ~<_  ( A  +c  ( A  u.  B ) ) )
75, 6syl 17 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  +c  B
)  ~<_  ( A  +c  ( A  u.  B
) ) )
8 cdacomen 7761 . . . 4  |-  ( A  +c  ( A  u.  B ) )  ~~  ( ( A  u.  B )  +c  A
)
9 domentr 6874 . . . 4  |-  ( ( ( A  +c  B
)  ~<_  ( A  +c  ( A  u.  B
) )  /\  ( A  +c  ( A  u.  B ) )  ~~  ( ( A  u.  B )  +c  A
) )  ->  ( A  +c  B )  ~<_  ( ( A  u.  B
)  +c  A ) )
107, 8, 9sylancl 646 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  +c  B
)  ~<_  ( ( A  u.  B )  +c  A ) )
11 simp3 962 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  om 
~<_  A )
12 ssun1 3299 . . . . . 6  |-  A  C_  ( A  u.  B
)
13 ssdomg 6861 . . . . . 6  |-  ( ( A  u.  B )  e.  dom  card  ->  ( A  C_  ( A  u.  B )  ->  A  ~<_  ( A  u.  B
) ) )
142, 12, 13ee10 1372 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  A  ~<_  ( A  u.  B ) )
15 domtr 6868 . . . . 5  |-  ( ( om  ~<_  A  /\  A  ~<_  ( A  u.  B
) )  ->  om  ~<_  ( A  u.  B ) )
1611, 14, 15syl2anc 645 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  om 
~<_  ( A  u.  B
) )
17 infcdaabs 7786 . . . 4  |-  ( ( ( A  u.  B
)  e.  dom  card  /\ 
om  ~<_  ( A  u.  B )  /\  A  ~<_  ( A  u.  B
) )  ->  (
( A  u.  B
)  +c  A ) 
~~  ( A  u.  B ) )
182, 16, 14, 17syl3anc 1187 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( A  u.  B )  +c  A
)  ~~  ( A  u.  B ) )
19 domentr 6874 . . 3  |-  ( ( ( A  +c  B
)  ~<_  ( ( A  u.  B )  +c  A )  /\  (
( A  u.  B
)  +c  A ) 
~~  ( A  u.  B ) )  -> 
( A  +c  B
)  ~<_  ( A  u.  B ) )
2010, 18, 19syl2anc 645 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  +c  B
)  ~<_  ( A  u.  B ) )
21 uncdadom 7751 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  u.  B )  ~<_  ( A  +c  B ) )
22213adant3 980 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  u.  B
)  ~<_  ( A  +c  B ) )
23 sbth 6935 . 2  |-  ( ( ( A  +c  B
)  ~<_  ( A  u.  B )  /\  ( A  u.  B )  ~<_  ( A  +c  B
) )  ->  ( A  +c  B )  ~~  ( A  u.  B
) )
2420, 22, 23syl2anc 645 1  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  +c  B
)  ~~  ( A  u.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ w3a 939    e. wcel 1621    u. cun 3111    C_ wss 3113   class class class wbr 3983   omcom 4614   dom cdm 4647  (class class class)co 5778    ~~ cen 6814    ~<_ cdom 6815   cardccrd 7522    +c ccda 7747
This theorem is referenced by:  alephadd  8153
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 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-oi 7179  df-card 7526  df-cda 7748
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