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Theorem rankelpr 7791
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 7790 . . . 4  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  ( A  u.  B ) )  e.  ( rank `  ( C  u.  D )
) )
61, 2rankun 7774 . . . 4  |-  ( rank `  ( A  u.  B
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) )
73, 4rankun 7774 . . . 4  |-  ( rank `  ( C  u.  D
) )  =  ( ( rank `  C
)  u.  ( rank `  D ) )
85, 6, 73eltr3g 2517 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  (
( rank `  A )  u.  ( rank `  B
) )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
9 rankon 7713 . . . . . 6  |-  ( rank `  C )  e.  On
10 rankon 7713 . . . . . 6  |-  ( rank `  D )  e.  On
119, 10onun2i 4689 . . . . 5  |-  ( (
rank `  C )  u.  ( rank `  D
) )  e.  On
1211onordi 4678 . . . 4  |-  Ord  (
( rank `  C )  u.  ( rank `  D
) )
13 ordsucelsuc 4794 . . . 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 7775 . 2  |-  ( rank `  { A ,  B } )  =  suc  ( ( rank `  A
)  u.  ( rank `  B ) )
173, 4rankpr 7775 . 2  |-  ( rank `  { C ,  D } )  =  suc  ( ( rank `  C
)  u.  ( rank `  D ) )
1815, 16, 173eltr4g 2518 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 1725   _Vcvv 2948    u. cun 3310   {cpr 3807   Ord word 4572   suc csuc 4575   ` cfv 5446   rankcrnk 7681
This theorem is referenced by:  rankelop  7792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-reg 7552  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660  df-r1 7682  df-rank 7683
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