MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankelun Structured version   Unicode version

Theorem rankelun 8193
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 8116 . . . . . 6  |-  ( rank `  C )  e.  On
2 rankon 8116 . . . . . 6  |-  ( rank `  D )  e.  On
31, 2onun2i 4945 . . . . 5  |-  ( (
rank `  C )  u.  ( rank `  D
) )  e.  On
43onordi 4934 . . . 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 3634 . . . 4  |-  ( (
rank `  A )  e.  ( rank `  C
)  ->  ( rank `  A )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
76adantr 465 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  A )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
8 elun2 3635 . . . 4  |-  ( (
rank `  B )  e.  ( rank `  D
)  ->  ( rank `  B )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
98adantl 466 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  B )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
10 ordunel 6551 . . 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 1219 . 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 8177 . 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 8177 . 2  |-  ( rank `  ( C  u.  D
) )  =  ( ( rank `  C
)  u.  ( rank `  D ) )
1811, 14, 173eltr4g 2560 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 369    e. wcel 1758   _Vcvv 3078    u. cun 3437   Ord word 4829   ` cfv 5529   rankcrnk 8084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-reg 7921  ax-inf2 7961
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-om 6590  df-recs 6945  df-rdg 6979  df-r1 8085  df-rank 8086
This theorem is referenced by:  rankelpr  8194  rankxplim  8200
  Copyright terms: Public domain W3C validator