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Theorem infcda 8077
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 8069 . . . . . . 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 3503 . . . . . 6  |-  B  C_  ( A  u.  B
)
4 ssdomg 7144 . . . . . 6  |-  ( ( A  u.  B )  e.  dom  card  ->  ( B  C_  ( A  u.  B )  ->  B  ~<_  ( A  u.  B
) ) )
52, 3, 4ee10 1385 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  B  ~<_  ( A  u.  B ) )
6 cdadom2 8056 . . . . 5  |-  ( B  ~<_  ( A  u.  B
)  ->  ( A  +c  B )  ~<_  ( A  +c  ( A  u.  B ) ) )
75, 6syl 16 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  +c  B
)  ~<_  ( A  +c  ( A  u.  B
) ) )
8 cdacomen 8050 . . . 4  |-  ( A  +c  ( A  u.  B ) )  ~~  ( ( A  u.  B )  +c  A
)
9 domentr 7157 . . . 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 644 . . 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 3502 . . . . . 6  |-  A  C_  ( A  u.  B
)
13 ssdomg 7144 . . . . . 6  |-  ( ( A  u.  B )  e.  dom  card  ->  ( A  C_  ( A  u.  B )  ->  A  ~<_  ( A  u.  B
) ) )
142, 12, 13ee10 1385 . . . . 5  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  A  ~<_  ( A  u.  B ) )
15 domtr 7151 . . . . 5  |-  ( ( om  ~<_  A  /\  A  ~<_  ( A  u.  B
) )  ->  om  ~<_  ( A  u.  B ) )
1611, 14, 15syl2anc 643 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  ->  om 
~<_  ( A  u.  B
) )
17 infcdaabs 8075 . . . 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 7157 . . 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 643 . 2  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  +c  B
)  ~<_  ( A  u.  B ) )
21 uncdadom 8040 . . 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 7218 . 2  |-  ( ( ( A  +c  B
)  ~<_  ( A  u.  B )  /\  ( A  u.  B )  ~<_  ( A  +c  B
) )  ->  ( A  +c  B )  ~~  ( A  u.  B
) )
2420, 22, 23syl2anc 643 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 936    e. wcel 1725    u. cun 3310    C_ wss 3312   class class class wbr 4204   omcom 4836   dom cdm 4869  (class class class)co 6072    ~~ cen 7097    ~<_ cdom 7098   cardccrd 7811    +c ccda 8036
This theorem is referenced by:  alephadd  8441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-oi 7468  df-card 7815  df-cda 8037
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