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Theorem infunsdom 8029
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 7073 . . 3  |-  ( A 
~<  B  ->  A  ~<_  B )
2 infunsdom1 8028 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  ( A  u.  B )  ~<  X )
32anass1rs 783 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  B  ~<  X )  /\  A  ~<_  B )  ->  ( A  u.  B )  ~<  X )
43adantlrl 701 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  /\  A  ~<_  B )  ->  ( A  u.  B )  ~<  X )
51, 4sylan2 461 . 2  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  /\  A  ~<  B )  ->  ( A  u.  B )  ~<  X )
6 simpll 731 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  X  e.  dom  card )
7 sdomdom 7073 . . . . . . 7  |-  ( B 
~<  X  ->  B  ~<_  X )
87ad2antll 710 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  B  ~<_  X )
9 numdom 7854 . . . . . 6  |-  ( ( X  e.  dom  card  /\  B  ~<_  X )  ->  B  e.  dom  card )
106, 8, 9syl2anc 643 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  B  e.  dom  card )
11 sdomdom 7073 . . . . . . 7  |-  ( A 
~<  X  ->  A  ~<_  X )
1211ad2antrl 709 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  A  ~<_  X )
13 numdom 7854 . . . . . 6  |-  ( ( X  e.  dom  card  /\  A  ~<_  X )  ->  A  e.  dom  card )
146, 12, 13syl2anc 643 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  A  e.  dom  card )
15 domtri2 7811 . . . . 5  |-  ( ( B  e.  dom  card  /\  A  e.  dom  card )  ->  ( B  ~<_  A  <->  -.  A  ~<  B ) )
1610, 14, 15syl2anc 643 . . . 4  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  ->  ( B  ~<_  A  <->  -.  A  ~<  B ) )
1716biimpar 472 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  /\  -.  A  ~<  B )  ->  B  ~<_  A )
18 uncom 3436 . . . . . 6  |-  ( A  u.  B )  =  ( B  u.  A
)
19 infunsdom1 8028 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( B  ~<_  A  /\  A  ~<  X ) )  ->  ( B  u.  A )  ~<  X )
2018, 19syl5eqbr 4188 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( B  ~<_  A  /\  A  ~<  X ) )  ->  ( A  u.  B )  ~<  X )
2120anass1rs 783 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  A  ~<  X )  /\  B  ~<_  A )  ->  ( A  u.  B )  ~<  X )
2221adantlrr 702 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  /\  B  ~<_  A )  ->  ( A  u.  B )  ~<  X )
2317, 22syldan 457 . 2  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<  X  /\  B  ~<  X ) )  /\  -.  A  ~<  B )  ->  ( A  u.  B )  ~<  X )
245, 23pm2.61dan 767 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 177    /\ wa 359    e. wcel 1717    u. cun 3263   class class class wbr 4155   omcom 4787   dom cdm 4820    ~<_ cdom 7045    ~< csdm 7046   cardccrd 7757
This theorem is referenced by:  csdfil  17849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-oi 7414  df-card 7761  df-cda 7983
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