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Theorem rankelun 7560
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 7483 . . . . . 6  |-  ( rank `  C )  e.  On
2 rankon 7483 . . . . . 6  |-  ( rank `  D )  e.  On
31, 2onun2i 4524 . . . . 5  |-  ( (
rank `  C )  u.  ( rank `  D
) )  e.  On
43onordi 4513 . . . 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 3355 . . . 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 3356 . . . 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 4634 . . 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 7544 . . 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 7544 . . 3  |-  ( rank `  ( C  u.  D
) )  =  ( ( rank `  C
)  u.  ( rank `  D ) )
1814, 17eleq12i 2361 . 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 1696   _Vcvv 2801    u. cun 3163   Ord word 4407   ` cfv 5271   rankcrnk 7451
This theorem is referenced by:  rankelpr  7561  rankxplim  7565
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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