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

Theorem rankelb 8023
Description: The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankelb  |-  ( B  e.  U. ( R1
" On )  -> 
( A  e.  B  ->  ( rank `  A
)  e.  ( rank `  B ) ) )

Proof of Theorem rankelb
StepHypRef Expression
1 r1elssi 8004 . . . . . 6  |-  ( B  e.  U. ( R1
" On )  ->  B  C_  U. ( R1
" On ) )
21sseld 3350 . . . . 5  |-  ( B  e.  U. ( R1
" On )  -> 
( A  e.  B  ->  A  e.  U. ( R1 " On ) ) )
3 rankidn 8021 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  -.  A  e.  ( R1 `  ( rank `  A
) ) )
42, 3syl6 33 . . . 4  |-  ( B  e.  U. ( R1
" On )  -> 
( A  e.  B  ->  -.  A  e.  ( R1 `  ( rank `  A ) ) ) )
54imp 429 . . 3  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  -.  A  e.  ( R1 `  ( rank `  A ) ) )
6 rankon 7994 . . . . 5  |-  ( rank `  B )  e.  On
7 rankon 7994 . . . . 5  |-  ( rank `  A )  e.  On
8 ontri1 4748 . . . . 5  |-  ( ( ( rank `  B
)  e.  On  /\  ( rank `  A )  e.  On )  ->  (
( rank `  B )  C_  ( rank `  A
)  <->  -.  ( rank `  A )  e.  (
rank `  B )
) )
96, 7, 8mp2an 672 . . . 4  |-  ( (
rank `  B )  C_  ( rank `  A
)  <->  -.  ( rank `  A )  e.  (
rank `  B )
)
10 rankdmr1 8000 . . . . . 6  |-  ( rank `  B )  e.  dom  R1
11 rankdmr1 8000 . . . . . 6  |-  ( rank `  A )  e.  dom  R1
12 r1ord3g 7978 . . . . . 6  |-  ( ( ( rank `  B
)  e.  dom  R1  /\  ( rank `  A
)  e.  dom  R1 )  ->  ( ( rank `  B )  C_  ( rank `  A )  -> 
( R1 `  ( rank `  B ) ) 
C_  ( R1 `  ( rank `  A )
) ) )
1310, 11, 12mp2an 672 . . . . 5  |-  ( (
rank `  B )  C_  ( rank `  A
)  ->  ( R1 `  ( rank `  B
) )  C_  ( R1 `  ( rank `  A
) ) )
14 r1rankidb 8003 . . . . . . 7  |-  ( B  e.  U. ( R1
" On )  ->  B  C_  ( R1 `  ( rank `  B )
) )
1514sselda 3351 . . . . . 6  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  A  e.  ( R1
`  ( rank `  B
) ) )
16 ssel 3345 . . . . . 6  |-  ( ( R1 `  ( rank `  B ) )  C_  ( R1 `  ( rank `  A ) )  -> 
( A  e.  ( R1 `  ( rank `  B ) )  ->  A  e.  ( R1 `  ( rank `  A
) ) ) )
1715, 16syl5com 30 . . . . 5  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  ( ( R1 `  ( rank `  B )
)  C_  ( R1 `  ( rank `  A
) )  ->  A  e.  ( R1 `  ( rank `  A ) ) ) )
1813, 17syl5 32 . . . 4  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  ( ( rank `  B
)  C_  ( rank `  A )  ->  A  e.  ( R1 `  ( rank `  A ) ) ) )
199, 18syl5bir 218 . . 3  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  ( -.  ( rank `  A )  e.  (
rank `  B )  ->  A  e.  ( R1
`  ( rank `  A
) ) ) )
205, 19mt3d 125 . 2  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  ( rank `  A
)  e.  ( rank `  B ) )
2120ex 434 1  |-  ( B  e.  U. ( R1
" On )  -> 
( A  e.  B  ->  ( rank `  A
)  e.  ( rank `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756    C_ wss 3323   U.cuni 4086   Oncon0 4714   dom cdm 4835   "cima 4838   ` cfv 5413   R1cr1 7961   rankcrnk 7962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-recs 6824  df-rdg 6858  df-r1 7963  df-rank 7964
This theorem is referenced by:  wfelirr  8024  rankval3b  8025  rankel  8038  rankunb  8049  rankuni2b  8052  rankcf  8936
  Copyright terms: Public domain W3C validator