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Theorem infcda 7829
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 7821 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  u.  B )  e.  dom  card )
213adant3 977 . . . . . 6  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  u.  B
)  e.  dom  card )
3 ssun2 3340 . . . . . 6  |-  B  C_  ( A  u.  B
)
4 ssdomg 6902 . . . . . 6  |-  ( ( A  u.  B )  e.  dom  card  ->  ( B  C_  ( A  u.  B )  ->  B  ~<_  ( A  u.  B
) ) )
52, 3, 4ee10 1368 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  B  ~<_  ( A  u.  B ) )
6 cdadom2 7808 . . . . 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 7802 . . . 4  |-  ( A  +c  ( A  u.  B ) )  ~~  ( ( A  u.  B )  +c  A
)
9 domentr 6915 . . . 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 645 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  +c  B
)  ~<_  ( ( A  u.  B )  +c  A ) )
11 simp3 959 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  om 
~<_  A )
12 ssun1 3339 . . . . . 6  |-  A  C_  ( A  u.  B
)
13 ssdomg 6902 . . . . . 6  |-  ( ( A  u.  B )  e.  dom  card  ->  ( A  C_  ( A  u.  B )  ->  A  ~<_  ( A  u.  B
) ) )
142, 12, 13ee10 1368 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  A  ~<_  ( A  u.  B ) )
15 domtr 6909 . . . . 5  |-  ( ( om  ~<_  A  /\  A  ~<_  ( A  u.  B
) )  ->  om  ~<_  ( A  u.  B ) )
1611, 14, 15syl2anc 644 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  om 
~<_  ( A  u.  B
) )
17 infcdaabs 7827 . . . 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 1184 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( A  u.  B )  +c  A
)  ~~  ( A  u.  B ) )
19 domentr 6915 . . 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 644 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  +c  B
)  ~<_  ( A  u.  B ) )
21 uncdadom 7792 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  u.  B )  ~<_  ( A  +c  B ) )
22213adant3 977 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  u.  B
)  ~<_  ( A  +c  B ) )
23 sbth 6976 . 2  |-  ( ( ( A  +c  B
)  ~<_  ( A  u.  B )  /\  ( A  u.  B )  ~<_  ( A  +c  B
) )  ->  ( A  +c  B )  ~~  ( A  u.  B
) )
2420, 22, 23syl2anc 644 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 936    e. wcel 1685    u. cun 3151    C_ wss 3153   class class class wbr 4024   omcom 4655   dom cdm 4688  (class class class)co 5819    ~~ cen 6855    ~<_ cdom 6856   cardccrd 7563    +c ccda 7788
This theorem is referenced by:  alephadd  8194
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-oi 7220  df-card 7567  df-cda 7789
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