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Theorem rankxplim2 7550
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 4454 . . . 4  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  (/)  e.  ( rank `  ( A  X.  B
) ) )
2 n0i 3460 . . . 4  |-  ( (/)  e.  ( rank `  ( A  X.  B ) )  ->  -.  ( rank `  ( A  X.  B
) )  =  (/) )
31, 2syl 15 . . 3  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  -.  ( rank `  ( A  X.  B
) )  =  (/) )
4 df-ne 2448 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  <->  -.  ( A  X.  B )  =  (/) )
5 rankxplim.1 . . . . . . 7  |-  A  e. 
_V
6 rankxplim.2 . . . . . . 7  |-  B  e. 
_V
75, 6xpex 4801 . . . . . 6  |-  ( A  X.  B )  e. 
_V
87rankeq0 7533 . . . . 5  |-  ( ( A  X.  B )  =  (/)  <->  ( rank `  ( A  X.  B ) )  =  (/) )
98notbii 287 . . . 4  |-  ( -.  ( A  X.  B
)  =  (/)  <->  -.  ( rank `  ( A  X.  B ) )  =  (/) )
104, 9bitr2i 241 . . 3  |-  ( -.  ( rank `  ( A  X.  B ) )  =  (/)  <->  ( A  X.  B )  =/=  (/) )
113, 10sylib 188 . 2  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  ( A  X.  B )  =/=  (/) )
12 limuni2 4453 . . . 4  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  U. ( rank `  ( A  X.  B ) ) )
13 limuni2 4453 . . . 4  |-  ( Lim  U. ( rank `  ( A  X.  B ) )  ->  Lim  U. U. ( rank `  ( A  X.  B ) ) )
1412, 13syl 15 . . 3  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  U. U. ( rank `  ( A  X.  B ) ) )
15 rankuni 7535 . . . . . 6  |-  ( rank `  U. U. ( A  X.  B ) )  =  U. ( rank `  U. ( A  X.  B ) )
16 rankuni 7535 . . . . . . 7  |-  ( rank `  U. ( A  X.  B ) )  = 
U. ( rank `  ( A  X.  B ) )
1716unieqi 3837 . . . . . 6  |-  U. ( rank `  U. ( A  X.  B ) )  =  U. U. ( rank `  ( A  X.  B ) )
1815, 17eqtr2i 2304 . . . . 5  |-  U. U. ( rank `  ( A  X.  B ) )  =  ( rank `  U. U. ( A  X.  B
) )
19 unixp 5205 . . . . . 6  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )
2019fveq2d 5529 . . . . 5  |-  ( ( A  X.  B )  =/=  (/)  ->  ( rank ` 
U. U. ( A  X.  B ) )  =  ( rank `  ( A  u.  B )
) )
2118, 20syl5eq 2327 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( rank `  ( A  X.  B ) )  =  ( rank `  ( A  u.  B )
) )
22 limeq 4404 . . . 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 15 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  ( Lim  U.
U. ( rank `  ( A  X.  B ) )  <->  Lim  ( rank `  ( A  u.  B )
) ) )
2414, 23syl5ib 210 . 2  |-  ( ( A  X.  B )  =/=  (/)  ->  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  ( rank `  ( A  u.  B )
) ) )
2511, 24mpcom 32 1  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  ( rank `  ( A  u.  B
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    u. cun 3150   (/)c0 3455   U.cuni 3827   Lim wlim 4393    X. cxp 4687   ` cfv 5255   rankcrnk 7435
This theorem is referenced by:  rankxpsuc  7552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437
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