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Theorem rankxplim2 7796
Description: If the rank of a cross product is a limit ordinal, so is the rank of the union of its arguments. (Contributed by NM, 19-Sep-2006.)
Hypotheses
Ref Expression
rankxplim.1  |-  A  e. 
_V
rankxplim.2  |-  B  e. 
_V
Assertion
Ref Expression
rankxplim2  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  ( rank `  ( A  u.  B
) ) )

Proof of Theorem rankxplim2
StepHypRef Expression
1 0ellim 4635 . . . 4  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  (/)  e.  ( rank `  ( A  X.  B
) ) )
2 n0i 3625 . . . 4  |-  ( (/)  e.  ( rank `  ( A  X.  B ) )  ->  -.  ( rank `  ( A  X.  B
) )  =  (/) )
31, 2syl 16 . . 3  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  -.  ( rank `  ( A  X.  B
) )  =  (/) )
4 df-ne 2600 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  <->  -.  ( A  X.  B )  =  (/) )
5 rankxplim.1 . . . . . . 7  |-  A  e. 
_V
6 rankxplim.2 . . . . . . 7  |-  B  e. 
_V
75, 6xpex 4982 . . . . . 6  |-  ( A  X.  B )  e. 
_V
87rankeq0 7779 . . . . 5  |-  ( ( A  X.  B )  =  (/)  <->  ( rank `  ( A  X.  B ) )  =  (/) )
98notbii 288 . . . 4  |-  ( -.  ( A  X.  B
)  =  (/)  <->  -.  ( rank `  ( A  X.  B ) )  =  (/) )
104, 9bitr2i 242 . . 3  |-  ( -.  ( rank `  ( A  X.  B ) )  =  (/)  <->  ( A  X.  B )  =/=  (/) )
113, 10sylib 189 . 2  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  ( A  X.  B )  =/=  (/) )
12 limuni2 4634 . . . 4  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  U. ( rank `  ( A  X.  B ) ) )
13 limuni2 4634 . . . 4  |-  ( Lim  U. ( rank `  ( A  X.  B ) )  ->  Lim  U. U. ( rank `  ( A  X.  B ) ) )
1412, 13syl 16 . . 3  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  U. U. ( rank `  ( A  X.  B ) ) )
15 rankuni 7781 . . . . . 6  |-  ( rank `  U. U. ( A  X.  B ) )  =  U. ( rank `  U. ( A  X.  B ) )
16 rankuni 7781 . . . . . . 7  |-  ( rank `  U. ( A  X.  B ) )  = 
U. ( rank `  ( A  X.  B ) )
1716unieqi 4017 . . . . . 6  |-  U. ( rank `  U. ( A  X.  B ) )  =  U. U. ( rank `  ( A  X.  B ) )
1815, 17eqtr2i 2456 . . . . 5  |-  U. U. ( rank `  ( A  X.  B ) )  =  ( rank `  U. U. ( A  X.  B
) )
19 unixp 5394 . . . . . 6  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )
2019fveq2d 5724 . . . . 5  |-  ( ( A  X.  B )  =/=  (/)  ->  ( rank ` 
U. U. ( A  X.  B ) )  =  ( rank `  ( A  u.  B )
) )
2118, 20syl5eq 2479 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( rank `  ( A  X.  B ) )  =  ( rank `  ( A  u.  B )
) )
22 limeq 4585 . . . 4  |-  ( U. U. ( rank `  ( A  X.  B ) )  =  ( rank `  ( A  u.  B )
)  ->  ( Lim  U.
U. ( rank `  ( A  X.  B ) )  <->  Lim  ( rank `  ( A  u.  B )
) ) )
2321, 22syl 16 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  ( Lim  U.
U. ( rank `  ( A  X.  B ) )  <->  Lim  ( rank `  ( A  u.  B )
) ) )
2414, 23syl5ib 211 . 2  |-  ( ( A  X.  B )  =/=  (/)  ->  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  ( rank `  ( A  u.  B )
) ) )
2511, 24mpcom 34 1  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  ( rank `  ( A  u.  B
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    u. cun 3310   (/)c0 3620   U.cuni 4007   Lim wlim 4574    X. cxp 4868   ` cfv 5446   rankcrnk 7681
This theorem is referenced by:  rankxpsuc  7798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-reg 7552  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660  df-r1 7682  df-rank 7683
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