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Theorem infunsdom1 8095
Description: The union of two sets that are strictly dominated by the infinite set  X is also dominated by  X. This version of infunsdom 8096 assumes additionally that  A is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
infunsdom1  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  ( A  u.  B )  ~<  X )

Proof of Theorem infunsdom1
StepHypRef Expression
1 simprl 734 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  A  ~<_  B )
2 domsdomtr 7244 . . . . 5  |-  ( ( A  ~<_  B  /\  B  ~<  om )  ->  A  ~<  om )
31, 2sylan 459 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  B  ~<  om )  ->  A  ~<  om )
4 unfi2 7378 . . . 4  |-  ( ( A  ~<  om  /\  B  ~<  om )  ->  ( A  u.  B )  ~<  om )
53, 4sylancom 650 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  B  ~<  om )  ->  ( A  u.  B )  ~<  om )
6 simpllr 737 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  B  ~<  om )  ->  om  ~<_  X )
7 sdomdomtr 7242 . . 3  |-  ( ( ( A  u.  B
)  ~<  om  /\  om  ~<_  X )  ->  ( A  u.  B )  ~<  X )
85, 6, 7syl2anc 644 . 2  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  B  ~<  om )  ->  ( A  u.  B )  ~<  X )
9 omelon 7603 . . . . . 6  |-  om  e.  On
10 onenon 7838 . . . . . 6  |-  ( om  e.  On  ->  om  e.  dom  card )
119, 10ax-mp 8 . . . . 5  |-  om  e.  dom  card
12 simpll 732 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  X  e.  dom  card )
13 sdomdom 7137 . . . . . . 7  |-  ( B 
~<  X  ->  B  ~<_  X )
1413ad2antll 711 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  B  ~<_  X )
15 numdom 7921 . . . . . 6  |-  ( ( X  e.  dom  card  /\  B  ~<_  X )  ->  B  e.  dom  card )
1612, 14, 15syl2anc 644 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  B  e.  dom  card )
17 domtri2 7878 . . . . 5  |-  ( ( om  e.  dom  card  /\  B  e.  dom  card )  ->  ( om  ~<_  B  <->  -.  B  ~<  om ) )
1811, 16, 17sylancr 646 . . . 4  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  ( om  ~<_  B  <->  -.  B  ~<  om ) )
1918biimpar 473 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  -.  B  ~<  om )  ->  om  ~<_  B )
20 uncom 3493 . . . . 5  |-  ( A  u.  B )  =  ( B  u.  A
)
2116adantr 453 . . . . . 6  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  B  e.  dom  card )
22 simpr 449 . . . . . 6  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  om  ~<_  B )
231adantr 453 . . . . . 6  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  A  ~<_  B )
24 infunabs 8089 . . . . . 6  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B  /\  A  ~<_  B )  ->  ( B  u.  A )  ~~  B )
2521, 22, 23, 24syl3anc 1185 . . . . 5  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  ( B  u.  A )  ~~  B
)
2620, 25syl5eqbr 4247 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  ( A  u.  B )  ~~  B
)
27 simplrr 739 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  B  ~<  X )
28 ensdomtr 7245 . . . 4  |-  ( ( ( A  u.  B
)  ~~  B  /\  B  ~<  X )  -> 
( A  u.  B
)  ~<  X )
2926, 27, 28syl2anc 644 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  ( A  u.  B )  ~<  X )
3019, 29syldan 458 . 2  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  -.  B  ~<  om )  ->  ( A  u.  B )  ~<  X )
318, 30pm2.61dan 768 1  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  ( A  u.  B )  ~<  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726    u. cun 3320   class class class wbr 4214   Oncon0 4583   omcom 4847   dom cdm 4880    ~~ cen 7108    ~<_ cdom 7109    ~< csdm 7110   cardccrd 7824
This theorem is referenced by:  infunsdom  8096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-oi 7481  df-card 7828  df-cda 8050
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