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Theorem infcda 7718
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 7710 . . . . . . 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 3249 . . . . . 6  |-  B  C_  ( A  u.  B
)
4 ssdomg 6793 . . . . . 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 7697 . . . . 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 7691 . . . 4  |-  ( A  +c  ( A  u.  B ) )  ~~  ( ( A  u.  B )  +c  A
)
9 domentr 6805 . . . 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 3248 . . . . . 6  |-  A  C_  ( A  u.  B
)
13 ssdomg 6793 . . . . . 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 6799 . . . . 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 7716 . . . 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 6805 . . 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 7681 . . 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 6866 . 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 3076    C_ wss 3078   class class class wbr 3920   omcom 4547   dom cdm 4580  (class class class)co 5710    ~~ cen 6746    ~<_ cdom 6747   cardccrd 7452    +c ccda 7677
This theorem is referenced by:  alephadd  8079
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-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
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 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-reu 2515  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-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  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-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-oi 7109  df-card 7456  df-cda 7678
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