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Theorem rankxplim2 7566
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 4470 . . . 4  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  (/)  e.  ( rank `  ( A  X.  B
) ) )
2 n0i 3473 . . . 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 2461 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  <->  -.  ( A  X.  B )  =  (/) )
5 rankxplim.1 . . . . . . 7  |-  A  e. 
_V
6 rankxplim.2 . . . . . . 7  |-  B  e. 
_V
75, 6xpex 4817 . . . . . 6  |-  ( A  X.  B )  e. 
_V
87rankeq0 7549 . . . . 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 4469 . . . 4  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  U. ( rank `  ( A  X.  B ) ) )
13 limuni2 4469 . . . 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 7551 . . . . . 6  |-  ( rank `  U. U. ( A  X.  B ) )  =  U. ( rank `  U. ( A  X.  B ) )
16 rankuni 7551 . . . . . . 7  |-  ( rank `  U. ( A  X.  B ) )  = 
U. ( rank `  ( A  X.  B ) )
1716unieqi 3853 . . . . . 6  |-  U. ( rank `  U. ( A  X.  B ) )  =  U. U. ( rank `  ( A  X.  B ) )
1815, 17eqtr2i 2317 . . . . 5  |-  U. U. ( rank `  ( A  X.  B ) )  =  ( rank `  U. U. ( A  X.  B
) )
19 unixp 5221 . . . . . 6  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )
2019fveq2d 5545 . . . . 5  |-  ( ( A  X.  B )  =/=  (/)  ->  ( rank ` 
U. U. ( A  X.  B ) )  =  ( rank `  ( A  u.  B )
) )
2118, 20syl5eq 2340 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( rank `  ( A  X.  B ) )  =  ( rank `  ( A  u.  B )
) )
22 limeq 4420 . . . 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 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    u. cun 3163   (/)c0 3468   U.cuni 3843   Lim wlim 4409    X. cxp 4703   ` cfv 5271   rankcrnk 7451
This theorem is referenced by:  rankxpsuc  7568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-r1 7452  df-rank 7453
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