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Theorem infcda 7850
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 7842 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  u.  B )  e.  dom  card )
213adant3 975 . . . . . 6  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  u.  B
)  e.  dom  card )
3 ssun2 3352 . . . . . 6  |-  B  C_  ( A  u.  B
)
4 ssdomg 6923 . . . . . 6  |-  ( ( A  u.  B )  e.  dom  card  ->  ( B  C_  ( A  u.  B )  ->  B  ~<_  ( A  u.  B
) ) )
52, 3, 4ee10 1366 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  B  ~<_  ( A  u.  B ) )
6 cdadom2 7829 . . . . 5  |-  ( B  ~<_  ( A  u.  B
)  ->  ( A  +c  B )  ~<_  ( A  +c  ( A  u.  B ) ) )
75, 6syl 15 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  +c  B
)  ~<_  ( A  +c  ( A  u.  B
) ) )
8 cdacomen 7823 . . . 4  |-  ( A  +c  ( A  u.  B ) )  ~~  ( ( A  u.  B )  +c  A
)
9 domentr 6936 . . . 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 643 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  +c  B
)  ~<_  ( ( A  u.  B )  +c  A ) )
11 simp3 957 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  om 
~<_  A )
12 ssun1 3351 . . . . . 6  |-  A  C_  ( A  u.  B
)
13 ssdomg 6923 . . . . . 6  |-  ( ( A  u.  B )  e.  dom  card  ->  ( A  C_  ( A  u.  B )  ->  A  ~<_  ( A  u.  B
) ) )
142, 12, 13ee10 1366 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  A  ~<_  ( A  u.  B ) )
15 domtr 6930 . . . . 5  |-  ( ( om  ~<_  A  /\  A  ~<_  ( A  u.  B
) )  ->  om  ~<_  ( A  u.  B ) )
1611, 14, 15syl2anc 642 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  om 
~<_  ( A  u.  B
) )
17 infcdaabs 7848 . . . 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 1182 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( A  u.  B )  +c  A
)  ~~  ( A  u.  B ) )
19 domentr 6936 . . 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 642 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  +c  B
)  ~<_  ( A  u.  B ) )
21 uncdadom 7813 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  u.  B )  ~<_  ( A  +c  B ) )
22213adant3 975 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  u.  B
)  ~<_  ( A  +c  B ) )
23 sbth 6997 . 2  |-  ( ( ( A  +c  B
)  ~<_  ( A  u.  B )  /\  ( A  u.  B )  ~<_  ( A  +c  B
) )  ->  ( A  +c  B )  ~~  ( A  u.  B
) )
2420, 22, 23syl2anc 642 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 4    /\ w3a 934    e. wcel 1696    u. cun 3163    C_ wss 3165   class class class wbr 4039   omcom 4672   dom cdm 4705  (class class class)co 5874    ~~ cen 6876    ~<_ cdom 6877   cardccrd 7584    +c ccda 7809
This theorem is referenced by:  alephadd  8215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-cda 7810
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