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Theorem rankelun 7544
Description: Rank membership is inherited by union. (Contributed by NM, 18-Sep-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2014.)
Hypotheses
Ref Expression
rankelun.1  |-  A  e. 
_V
rankelun.2  |-  B  e. 
_V
rankelun.3  |-  C  e. 
_V
rankelun.4  |-  D  e. 
_V
Assertion
Ref Expression
rankelun  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  ( A  u.  B ) )  e.  ( rank `  ( C  u.  D )
) )

Proof of Theorem rankelun
StepHypRef Expression
1 rankon 7467 . . . . . 6  |-  ( rank `  C )  e.  On
2 rankon 7467 . . . . . 6  |-  ( rank `  D )  e.  On
31, 2onun2i 4508 . . . . 5  |-  ( (
rank `  C )  u.  ( rank `  D
) )  e.  On
43onordi 4497 . . . 4  |-  Ord  (
( rank `  C )  u.  ( rank `  D
) )
54a1i 10 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  Ord  ( ( rank `  C
)  u.  ( rank `  D ) ) )
6 elun1 3342 . . . 4  |-  ( (
rank `  A )  e.  ( rank `  C
)  ->  ( rank `  A )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
76adantr 451 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  A )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
8 elun2 3343 . . . 4  |-  ( (
rank `  B )  e.  ( rank `  D
)  ->  ( rank `  B )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
98adantl 452 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  B )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
10 ordunel 4618 . . 3  |-  ( ( Ord  ( ( rank `  C )  u.  ( rank `  D ) )  /\  ( rank `  A
)  e.  ( (
rank `  C )  u.  ( rank `  D
) )  /\  ( rank `  B )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )  ->  ( ( rank `  A )  u.  ( rank `  B ) )  e.  ( ( rank `  C )  u.  ( rank `  D ) ) )
115, 7, 9, 10syl3anc 1182 . 2  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  (
( rank `  A )  u.  ( rank `  B
) )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
12 rankelun.1 . . . 4  |-  A  e. 
_V
13 rankelun.2 . . . 4  |-  B  e. 
_V
1412, 13rankun 7528 . . 3  |-  ( rank `  ( A  u.  B
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) )
15 rankelun.3 . . . 4  |-  C  e. 
_V
16 rankelun.4 . . . 4  |-  D  e. 
_V
1715, 16rankun 7528 . . 3  |-  ( rank `  ( C  u.  D
) )  =  ( ( rank `  C
)  u.  ( rank `  D ) )
1814, 17eleq12i 2348 . 2  |-  ( (
rank `  ( A  u.  B ) )  e.  ( rank `  ( C  u.  D )
)  <->  ( ( rank `  A )  u.  ( rank `  B ) )  e.  ( ( rank `  C )  u.  ( rank `  D ) ) )
1911, 18sylibr 203 1  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  ( A  u.  B ) )  e.  ( rank `  ( C  u.  D )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   _Vcvv 2788    u. cun 3150   Ord word 4391   ` cfv 5255   rankcrnk 7435
This theorem is referenced by:  rankelpr  7545  rankxplim  7549
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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