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Theorem rankxplim2 8348
Description: If the rank of a Cartesian 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 5484 . . . 4  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  (/)  e.  ( rank `  ( A  X.  B
) ) )
2 n0i 3735 . . . 4  |-  ( (/)  e.  ( rank `  ( A  X.  B ) )  ->  -.  ( rank `  ( A  X.  B
) )  =  (/) )
31, 2syl 17 . . 3  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  -.  ( rank `  ( A  X.  B
) )  =  (/) )
4 df-ne 2623 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  <->  -.  ( A  X.  B )  =  (/) )
5 rankxplim.1 . . . . . . 7  |-  A  e. 
_V
6 rankxplim.2 . . . . . . 7  |-  B  e. 
_V
75, 6xpex 6592 . . . . . 6  |-  ( A  X.  B )  e. 
_V
87rankeq0 8329 . . . . 5  |-  ( ( A  X.  B )  =  (/)  <->  ( rank `  ( A  X.  B ) )  =  (/) )
98notbii 298 . . . 4  |-  ( -.  ( A  X.  B
)  =  (/)  <->  -.  ( rank `  ( A  X.  B ) )  =  (/) )
104, 9bitr2i 254 . . 3  |-  ( -.  ( rank `  ( A  X.  B ) )  =  (/)  <->  ( A  X.  B )  =/=  (/) )
113, 10sylib 200 . 2  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  ( A  X.  B )  =/=  (/) )
12 limuni2 5483 . . . 4  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  U. ( rank `  ( A  X.  B ) ) )
13 limuni2 5483 . . . 4  |-  ( Lim  U. ( rank `  ( A  X.  B ) )  ->  Lim  U. U. ( rank `  ( A  X.  B ) ) )
1412, 13syl 17 . . 3  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  U. U. ( rank `  ( A  X.  B ) ) )
15 rankuni 8331 . . . . . 6  |-  ( rank `  U. U. ( A  X.  B ) )  =  U. ( rank `  U. ( A  X.  B ) )
16 rankuni 8331 . . . . . . 7  |-  ( rank `  U. ( A  X.  B ) )  = 
U. ( rank `  ( A  X.  B ) )
1716unieqi 4206 . . . . . 6  |-  U. ( rank `  U. ( A  X.  B ) )  =  U. U. ( rank `  ( A  X.  B ) )
1815, 17eqtr2i 2473 . . . . 5  |-  U. U. ( rank `  ( A  X.  B ) )  =  ( rank `  U. U. ( A  X.  B
) )
19 unixp 5368 . . . . . 6  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )
2019fveq2d 5867 . . . . 5  |-  ( ( A  X.  B )  =/=  (/)  ->  ( rank ` 
U. U. ( A  X.  B ) )  =  ( rank `  ( A  u.  B )
) )
2118, 20syl5eq 2496 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( rank `  ( A  X.  B ) )  =  ( rank `  ( A  u.  B )
) )
22 limeq 5434 . . . 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 17 . . 3  |-  ( ( A  X.  B )  =/=  (/)  ->  ( Lim  U.
U. ( rank `  ( A  X.  B ) )  <->  Lim  ( rank `  ( A  u.  B )
) ) )
2414, 23syl5ib 223 . 2  |-  ( ( A  X.  B )  =/=  (/)  ->  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  ( rank `  ( A  u.  B )
) ) )
2511, 24mpcom 37 1  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  ( rank `  ( A  u.  B
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    = wceq 1443    e. wcel 1886    =/= wne 2621   _Vcvv 3044    u. cun 3401   (/)c0 3730   U.cuni 4197    X. cxp 4831   Lim wlim 5423   ` cfv 5581   rankcrnk 8231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-reg 8104  ax-inf2 8143
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6690  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-r1 8232  df-rank 8233
This theorem is referenced by:  rankxpsuc  8350
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