MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infunsdom1 Unicode version

Theorem infunsdom1 7884
Description: The union of two sets that are strictly dominated by the infinite set  X is also dominated by  X. This version of infunsdom 7885 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 732 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  A  ~<_  B )
2 domsdomtr 7039 . . . . 5  |-  ( ( A  ~<_  B  /\  B  ~<  om )  ->  A  ~<  om )
31, 2sylan 457 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  B  ~<  om )  ->  A  ~<  om )
4 unfi2 7171 . . . 4  |-  ( ( A  ~<  om  /\  B  ~<  om )  ->  ( A  u.  B )  ~<  om )
53, 4sylancom 648 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  B  ~<  om )  ->  ( A  u.  B )  ~<  om )
6 simpllr 735 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  B  ~<  om )  ->  om  ~<_  X )
7 sdomdomtr 7037 . . 3  |-  ( ( ( A  u.  B
)  ~<  om  /\  om  ~<_  X )  ->  ( A  u.  B )  ~<  X )
85, 6, 7syl2anc 642 . 2  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  B  ~<  om )  ->  ( A  u.  B )  ~<  X )
9 omelon 7392 . . . . . 6  |-  om  e.  On
10 onenon 7627 . . . . . 6  |-  ( om  e.  On  ->  om  e.  dom  card )
119, 10ax-mp 8 . . . . 5  |-  om  e.  dom  card
12 simpll 730 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  X  e.  dom  card )
13 sdomdom 6932 . . . . . . 7  |-  ( B 
~<  X  ->  B  ~<_  X )
1413ad2antll 709 . . . . . 6  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  B  ~<_  X )
15 numdom 7710 . . . . . 6  |-  ( ( X  e.  dom  card  /\  B  ~<_  X )  ->  B  e.  dom  card )
1612, 14, 15syl2anc 642 . . . . 5  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  B  e.  dom  card )
17 domtri2 7667 . . . . 5  |-  ( ( om  e.  dom  card  /\  B  e.  dom  card )  ->  ( om  ~<_  B  <->  -.  B  ~<  om ) )
1811, 16, 17sylancr 644 . . . 4  |-  ( ( ( X  e.  dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  ->  ( om  ~<_  B  <->  -.  B  ~<  om ) )
1918biimpar 471 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  -.  B  ~<  om )  ->  om  ~<_  B )
20 uncom 3353 . . . . 5  |-  ( A  u.  B )  =  ( B  u.  A
)
2116adantr 451 . . . . . 6  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  B  e.  dom  card )
22 simpr 447 . . . . . 6  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  om  ~<_  B )
231adantr 451 . . . . . 6  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  A  ~<_  B )
24 infunabs 7878 . . . . . 6  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B  /\  A  ~<_  B )  ->  ( B  u.  A )  ~~  B )
2521, 22, 23, 24syl3anc 1182 . . . . 5  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  ( B  u.  A )  ~~  B
)
2620, 25syl5eqbr 4093 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  ( A  u.  B )  ~~  B
)
27 simplrr 737 . . . 4  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  B  ~<  X )
28 ensdomtr 7040 . . . 4  |-  ( ( ( A  u.  B
)  ~~  B  /\  B  ~<  X )  -> 
( A  u.  B
)  ~<  X )
2926, 27, 28syl2anc 642 . . 3  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  om  ~<_  B )  ->  ( A  u.  B )  ~<  X )
3019, 29syldan 456 . 2  |-  ( ( ( ( X  e. 
dom  card  /\  om  ~<_  X )  /\  ( A  ~<_  B  /\  B  ~<  X ) )  /\  -.  B  ~<  om )  ->  ( A  u.  B )  ~<  X )
318, 30pm2.61dan 766 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 176    /\ wa 358    e. wcel 1701    u. cun 3184   class class class wbr 4060   Oncon0 4429   omcom 4693   dom cdm 4726    ~~ cen 6903    ~<_ cdom 6904    ~< csdm 6905   cardccrd 7613
This theorem is referenced by:  infunsdom  7885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-oi 7270  df-card 7617  df-cda 7839
  Copyright terms: Public domain W3C validator