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Theorem infunsdom1 8049
Description: The union of two sets that are strictly dominated by the infinite set  X is also dominated by  X. This version of infunsdom 8050 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 733 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  A  ~<_  B )
2 domsdomtr 7201 . . . . 5  |-  ( ( A  ~<_  B  /\  B  ~<  om )  ->  A  ~<  om )
31, 2sylan 458 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  B  ~<  om )  ->  A  ~<  om )
4 unfi2 7335 . . . 4  |-  ( ( A  ~<  om  /\  B  ~<  om )  ->  ( A  u.  B )  ~<  om )
53, 4sylancom 649 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  B  ~<  om )  ->  ( A  u.  B )  ~<  om )
6 simpllr 736 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  B  ~<  om )  ->  om  ~<_  X )
7 sdomdomtr 7199 . . 3  |-  ( ( ( A  u.  B
)  ~<  om  /\  om  ~<_  X )  ->  ( A  u.  B )  ~<  X )
85, 6, 7syl2anc 643 . 2  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  B  ~<  om )  ->  ( A  u.  B )  ~<  X )
9 omelon 7557 . . . . . 6  |-  om  e.  On
10 onenon 7792 . . . . . 6  |-  ( om  e.  On  ->  om  e.  dom  card )
119, 10ax-mp 8 . . . . 5  |-  om  e.  dom  card
12 simpll 731 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  X  e.  dom  card )
13 sdomdom 7094 . . . . . . 7  |-  ( B 
~<  X  ->  B  ~<_  X )
1413ad2antll 710 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  B  ~<_  X )
15 numdom 7875 . . . . . 6  |-  ( ( X  e.  dom  card  /\  B  ~<_  X )  ->  B  e.  dom  card )
1612, 14, 15syl2anc 643 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  B  e.  dom  card )
17 domtri2 7832 . . . . 5  |-  ( ( om  e.  dom  card  /\  B  e.  dom  card )  ->  ( om  ~<_  B  <->  -.  B  ~<  om ) )
1811, 16, 17sylancr 645 . . . 4  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  ( om  ~<_  B  <->  -.  B  ~<  om ) )
1918biimpar 472 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  -.  B  ~<  om )  ->  om  ~<_  B )
20 uncom 3451 . . . . 5  |-  ( A  u.  B )  =  ( B  u.  A
)
2116adantr 452 . . . . . 6  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  B  e.  dom  card )
22 simpr 448 . . . . . 6  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  om  ~<_  B )
231adantr 452 . . . . . 6  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  A  ~<_  B )
24 infunabs 8043 . . . . . 6  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B  /\  A  ~<_  B )  ->  ( B  u.  A )  ~~  B )
2521, 22, 23, 24syl3anc 1184 . . . . 5  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  ( B  u.  A )  ~~  B
)
2620, 25syl5eqbr 4205 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  ( A  u.  B )  ~~  B
)
27 simplrr 738 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  B  ~<  X )
28 ensdomtr 7202 . . . 4  |-  ( ( ( A  u.  B
)  ~~  B  /\  B  ~<  X )  -> 
( A  u.  B
)  ~<  X )
2926, 27, 28syl2anc 643 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  ( A  u.  B )  ~<  X )
3019, 29syldan 457 . 2  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  -.  B  ~<  om )  ->  ( A  u.  B )  ~<  X )
318, 30pm2.61dan 767 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 177    /\ wa 359    e. wcel 1721    u. cun 3278   class class class wbr 4172   Oncon0 4541   omcom 4804   dom cdm 4837    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067   cardccrd 7778
This theorem is referenced by:  infunsdom  8050
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-cda 8004
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