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Theorem infunsdom 7840
Description: The union of two sets that are strictly dominated by the infinite set  X is also strictly dominated by  X. (Contributed by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
infunsdom  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  ( A  u.  B )  ~<  X )

Proof of Theorem infunsdom
StepHypRef Expression
1 sdomdom 6889 . . 3  |-  ( A 
~<  B  ->  A  ~<_  B )
2 infunsdom1 7839 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  ( A  u.  B )  ~<  X )
32anass1rs 782 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  B  ~<  X )  /\  A  ~<_  B )  ->  ( A  u.  B )  ~<  X )
43adantlrl 700 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  /\  A  ~<_  B )  ->  ( A  u.  B )  ~<  X )
51, 4sylan2 460 . 2  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  /\  A  ~<  B )  ->  ( A  u.  B )  ~<  X )
6 simpll 730 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  X  e.  dom  card )
7 sdomdom 6889 . . . . . . 7  |-  ( B 
~<  X  ->  B  ~<_  X )
87ad2antll 709 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  B  ~<_  X )
9 numdom 7665 . . . . . 6  |-  ( ( X  e.  dom  card  /\  B  ~<_  X )  ->  B  e.  dom  card )
106, 8, 9syl2anc 642 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  B  e.  dom  card )
11 sdomdom 6889 . . . . . . 7  |-  ( A 
~<  X  ->  A  ~<_  X )
1211ad2antrl 708 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  A  ~<_  X )
13 numdom 7665 . . . . . 6  |-  ( ( X  e.  dom  card  /\  A  ~<_  X )  ->  A  e.  dom  card )
146, 12, 13syl2anc 642 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  A  e.  dom  card )
15 domtri2 7622 . . . . 5  |-  ( ( B  e.  dom  card  /\  A  e.  dom  card )  ->  ( B  ~<_  A  <->  -.  A  ~<  B ) )
1610, 14, 15syl2anc 642 . . . 4  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  ( B  ~<_  A  <->  -.  A  ~<  B ) )
1716biimpar 471 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  /\  -.  A  ~<  B )  ->  B  ~<_  A )
18 uncom 3319 . . . . . 6  |-  ( A  u.  B )  =  ( B  u.  A
)
19 infunsdom1 7839 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( B  ~<_  A  /\  A  ~<  X ) )  ->  ( B  u.  A )  ~<  X )
2018, 19syl5eqbr 4056 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( B  ~<_  A  /\  A  ~<  X ) )  ->  ( A  u.  B )  ~<  X )
2120anass1rs 782 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  A  ~<  X )  /\  B  ~<_  A )  ->  ( A  u.  B )  ~<  X )
2221adantlrr 701 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  /\  B  ~<_  A )  ->  ( A  u.  B )  ~<  X )
2317, 22syldan 456 . 2  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  /\  -.  A  ~<  B )  ->  ( A  u.  B )  ~<  X )
245, 23pm2.61dan 766 1  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  ( A  u.  B )  ~<  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    u. cun 3150   class class class wbr 4023   omcom 4656   dom cdm 4689    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568
This theorem is referenced by:  csdfil  17589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-cda 7794
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