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Theorem rankelb 8240
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 8221 . . . . . 6  |-  ( B  e.  U. ( R1
" On )  ->  B  C_  U. ( R1
" On ) )
21sseld 3485 . . . . 5  |-  ( B  e.  U. ( R1
" On )  -> 
( A  e.  B  ->  A  e.  U. ( R1 " On ) ) )
3 rankidn 8238 . . . . 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 8211 . . . . 5  |-  ( rank `  B )  e.  On
7 rankon 8211 . . . . 5  |-  ( rank `  A )  e.  On
8 ontri1 4898 . . . . 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 8217 . . . . . 6  |-  ( rank `  B )  e.  dom  R1
11 rankdmr1 8217 . . . . . 6  |-  ( rank `  A )  e.  dom  R1
12 r1ord3g 8195 . . . . . 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 8220 . . . . . . 7  |-  ( B  e.  U. ( R1
" On )  ->  B  C_  ( R1 `  ( rank `  B )
) )
1514sselda 3486 . . . . . 6  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  A  e.  ( R1
`  ( rank `  B
) ) )
16 ssel 3480 . . . . . 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 1802    C_ wss 3458   U.cuni 4230   Oncon0 4864   dom cdm 4985   "cima 4988   ` cfv 5574   R1cr1 8178   rankcrnk 8179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-om 6682  df-recs 7040  df-rdg 7074  df-r1 8180  df-rank 8181
This theorem is referenced by:  wfelirr  8241  rankval3b  8242  rankel  8255  rankunb  8266  rankuni2b  8269  rankcf  9153
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