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Theorem rankxplim2 8354
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 5502 . . . 4  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  (/)  e.  ( rank `  ( A  X.  B
) ) )
2 n0i 3767 . . . 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 2621 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  <->  -.  ( A  X.  B )  =  (/) )
5 rankxplim.1 . . . . . . 7  |-  A  e. 
_V
6 rankxplim.2 . . . . . . 7  |-  B  e. 
_V
75, 6xpex 6607 . . . . . 6  |-  ( A  X.  B )  e. 
_V
87rankeq0 8335 . . . . 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 5501 . . . 4  |-  ( Lim  ( rank `  ( A  X.  B ) )  ->  Lim  U. ( rank `  ( A  X.  B ) ) )
13 limuni2 5501 . . . 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 8337 . . . . . 6  |-  ( rank `  U. U. ( A  X.  B ) )  =  U. ( rank `  U. ( A  X.  B ) )
16 rankuni 8337 . . . . . . 7  |-  ( rank `  U. ( A  X.  B ) )  = 
U. ( rank `  ( A  X.  B ) )
1716unieqi 4226 . . . . . 6  |-  U. ( rank `  U. ( A  X.  B ) )  =  U. U. ( rank `  ( A  X.  B ) )
1815, 17eqtr2i 2453 . . . . 5  |-  U. U. ( rank `  ( A  X.  B ) )  =  ( rank `  U. U. ( A  X.  B
) )
19 unixp 5386 . . . . . 6  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( A  X.  B )  =  ( A  u.  B
) )
2019fveq2d 5883 . . . . 5  |-  ( ( A  X.  B )  =/=  (/)  ->  ( rank ` 
U. U. ( A  X.  B ) )  =  ( rank `  ( A  u.  B )
) )
2118, 20syl5eq 2476 . . . 4  |-  ( ( A  X.  B )  =/=  (/)  ->  U. U. ( rank `  ( A  X.  B ) )  =  ( rank `  ( A  u.  B )
) )
22 limeq 5452 . . . 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 38 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 1438    e. wcel 1869    =/= wne 2619   _Vcvv 3082    u. cun 3435   (/)c0 3762   U.cuni 4217    X. cxp 4849   Lim wlim 5441   ` cfv 5599   rankcrnk 8237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-reg 8111  ax-inf2 8150
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-om 6705  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-r1 8238  df-rank 8239
This theorem is referenced by:  rankxpsuc  8356
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