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Theorem rankelun 7800
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 7723 . . . . . 6  |-  ( rank `  C )  e.  On
2 rankon 7723 . . . . . 6  |-  ( rank `  D )  e.  On
31, 2onun2i 4699 . . . . 5  |-  ( (
rank `  C )  u.  ( rank `  D
) )  e.  On
43onordi 4688 . . . 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 3516 . . . 4  |-  ( (
rank `  A )  e.  ( rank `  C
)  ->  ( rank `  A )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
76adantr 453 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  A )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
8 elun2 3517 . . . 4  |-  ( (
rank `  B )  e.  ( rank `  D
)  ->  ( rank `  B )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
98adantl 454 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  B )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
10 ordunel 4809 . . 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 1185 . 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 7784 . 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 7784 . 2  |-  ( rank `  ( C  u.  D
) )  =  ( ( rank `  C
)  u.  ( rank `  D ) )
1811, 14, 173eltr4g 2521 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 360    e. wcel 1726   _Vcvv 2958    u. cun 3320   Ord word 4582   ` cfv 5456   rankcrnk 7691
This theorem is referenced by:  rankelpr  7801  rankxplim  7805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-reg 7562  ax-inf2 7598
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-recs 6635  df-rdg 6670  df-r1 7692  df-rank 7693
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