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Theorem rankelpr 7732
Description: Rank membership is inherited by unordered pairs. (Contributed by NM, 18-Sep-2006.) (Revised 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
rankelpr  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  { A ,  B } )  e.  (
rank `  { C ,  D } ) )

Proof of Theorem rankelpr
StepHypRef Expression
1 rankelun.1 . . . . 5  |-  A  e. 
_V
2 rankelun.2 . . . . 5  |-  B  e. 
_V
3 rankelun.3 . . . . 5  |-  C  e. 
_V
4 rankelun.4 . . . . 5  |-  D  e. 
_V
51, 2, 3, 4rankelun 7731 . . . 4  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  ( A  u.  B ) )  e.  ( rank `  ( C  u.  D )
) )
61, 2rankun 7715 . . . 4  |-  ( rank `  ( A  u.  B
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) )
73, 4rankun 7715 . . . 4  |-  ( rank `  ( C  u.  D
) )  =  ( ( rank `  C
)  u.  ( rank `  D ) )
85, 6, 73eltr3g 2469 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  (
( rank `  A )  u.  ( rank `  B
) )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
9 rankon 7654 . . . . . 6  |-  ( rank `  C )  e.  On
10 rankon 7654 . . . . . 6  |-  ( rank `  D )  e.  On
119, 10onun2i 4637 . . . . 5  |-  ( (
rank `  C )  u.  ( rank `  D
) )  e.  On
1211onordi 4626 . . . 4  |-  Ord  (
( rank `  C )  u.  ( rank `  D
) )
13 ordsucelsuc 4742 . . . 4  |-  ( Ord  ( ( rank `  C
)  u.  ( rank `  D ) )  -> 
( ( ( rank `  A )  u.  ( rank `  B ) )  e.  ( ( rank `  C )  u.  ( rank `  D ) )  <->  suc  ( ( rank `  A
)  u.  ( rank `  B ) )  e. 
suc  ( ( rank `  C )  u.  ( rank `  D ) ) ) )
1412, 13ax-mp 8 . . 3  |-  ( ( ( rank `  A
)  u.  ( rank `  B ) )  e.  ( ( rank `  C
)  u.  ( rank `  D ) )  <->  suc  ( (
rank `  A )  u.  ( rank `  B
) )  e.  suc  ( ( rank `  C
)  u.  ( rank `  D ) ) )
158, 14sylib 189 . 2  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  suc  ( ( rank `  A
)  u.  ( rank `  B ) )  e. 
suc  ( ( rank `  C )  u.  ( rank `  D ) ) )
161, 2rankpr 7716 . 2  |-  ( rank `  { A ,  B } )  =  suc  ( ( rank `  A
)  u.  ( rank `  B ) )
173, 4rankpr 7716 . 2  |-  ( rank `  { C ,  D } )  =  suc  ( ( rank `  C
)  u.  ( rank `  D ) )
1815, 16, 173eltr4g 2470 1  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  { A ,  B } )  e.  (
rank `  { C ,  D } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717   _Vcvv 2899    u. cun 3261   {cpr 3758   Ord word 4521   suc csuc 4524   ` cfv 5394   rankcrnk 7622
This theorem is referenced by:  rankelop  7733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-reg 7493  ax-inf2 7529
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-recs 6569  df-rdg 6604  df-r1 7623  df-rank 7624
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