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Theorem rankelun 8225
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 8148 . . . . . 6  |-  ( rank `  C )  e.  On
2 rankon 8148 . . . . . 6  |-  ( rank `  D )  e.  On
31, 2onun2i 4924 . . . . 5  |-  ( (
rank `  C )  u.  ( rank `  D
) )  e.  On
43onordi 4913 . . . 4  |-  Ord  (
( rank `  C )  u.  ( rank `  D
) )
54a1i 11 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  Ord  ( ( rank `  C
)  u.  ( rank `  D ) ) )
6 elun1 3602 . . . 4  |-  ( (
rank `  A )  e.  ( rank `  C
)  ->  ( rank `  A )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
76adantr 463 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  A )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
8 elun2 3603 . . . 4  |-  ( (
rank `  B )  e.  ( rank `  D
)  ->  ( rank `  B )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
98adantl 464 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  B )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
10 ordunel 6583 . . 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 1226 . 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 . . 3  |-  A  e. 
_V
13 rankelun.2 . . 3  |-  B  e. 
_V
1412, 13rankun 8209 . 2  |-  ( rank `  ( A  u.  B
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) )
15 rankelun.3 . . 3  |-  C  e. 
_V
16 rankelun.4 . . 3  |-  D  e. 
_V
1715, 16rankun 8209 . 2  |-  ( rank `  ( C  u.  D
) )  =  ( ( rank `  C
)  u.  ( rank `  D ) )
1811, 14, 173eltr4g 2502 1  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  ( A  u.  B ) )  e.  ( rank `  ( C  u.  D )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1836   _Vcvv 3051    u. cun 3404   Ord word 4808   ` cfv 5513   rankcrnk 8116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-reg 7955  ax-inf2 7994
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-om 6622  df-recs 6982  df-rdg 7016  df-r1 8117  df-rank 8118
This theorem is referenced by:  rankelpr  8226  rankxplim  8232
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