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Theorem rankelpr 8294
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 8293 . . . 4  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  ( A  u.  B ) )  e.  ( rank `  ( C  u.  D )
) )
61, 2rankun 8277 . . . 4  |-  ( rank `  ( A  u.  B
) )  =  ( ( rank `  A
)  u.  ( rank `  B ) )
73, 4rankun 8277 . . . 4  |-  ( rank `  ( C  u.  D
) )  =  ( ( rank `  C
)  u.  ( rank `  D ) )
85, 6, 73eltr3g 2547 . . 3  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  (
( rank `  A )  u.  ( rank `  B
) )  e.  ( ( rank `  C
)  u.  ( rank `  D ) ) )
9 rankon 8216 . . . . . 6  |-  ( rank `  C )  e.  On
10 rankon 8216 . . . . . 6  |-  ( rank `  D )  e.  On
119, 10onun2i 4983 . . . . 5  |-  ( (
rank `  C )  u.  ( rank `  D
) )  e.  On
1211onordi 4972 . . . 4  |-  Ord  (
( rank `  C )  u.  ( rank `  D
) )
13 ordsucelsuc 6642 . . . 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 5 . . 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 196 . 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 8278 . 2  |-  ( rank `  { A ,  B } )  =  suc  ( ( rank `  A
)  u.  ( rank `  B ) )
173, 4rankpr 8278 . 2  |-  ( rank `  { C ,  D } )  =  suc  ( ( rank `  C
)  u.  ( rank `  D ) )
1815, 16, 173eltr4g 2549 1  |-  ( ( ( rank `  A
)  e.  ( rank `  C )  /\  ( rank `  B )  e.  ( rank `  D
) )  ->  ( rank `  { A ,  B } )  e.  (
rank `  { C ,  D } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1804   _Vcvv 3095    u. cun 3459   {cpr 4016   Ord word 4867   suc csuc 4870   ` cfv 5578   rankcrnk 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-reg 8021  ax-inf2 8061
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-recs 7044  df-rdg 7078  df-r1 8185  df-rank 8186
This theorem is referenced by:  rankelop  8295
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