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Theorem rankelb 7492
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 7473 . . . . . 6  |-  ( B  e.  U. ( R1
" On )  ->  B  C_  U. ( R1
" On ) )
21sseld 3180 . . . . 5  |-  ( B  e.  U. ( R1
" On )  -> 
( A  e.  B  ->  A  e.  U. ( R1 " On ) ) )
3 rankidn 7490 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  -.  A  e.  ( R1 `  ( rank `  A
) ) )
42, 3syl6 29 . . . 4  |-  ( B  e.  U. ( R1
" On )  -> 
( A  e.  B  ->  -.  A  e.  ( R1 `  ( rank `  A ) ) ) )
54imp 418 . . 3  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  -.  A  e.  ( R1 `  ( rank `  A ) ) )
6 rankon 7463 . . . . 5  |-  ( rank `  B )  e.  On
7 rankon 7463 . . . . 5  |-  ( rank `  A )  e.  On
8 ontri1 4425 . . . . 5  |-  ( ( ( rank `  B
)  e.  On  /\  ( rank `  A )  e.  On )  ->  (
( rank `  B )  C_  ( rank `  A
)  <->  -.  ( rank `  A )  e.  (
rank `  B )
) )
96, 7, 8mp2an 653 . . . 4  |-  ( (
rank `  B )  C_  ( rank `  A
)  <->  -.  ( rank `  A )  e.  (
rank `  B )
)
10 rankdmr1 7469 . . . . . 6  |-  ( rank `  B )  e.  dom  R1
11 rankdmr1 7469 . . . . . 6  |-  ( rank `  A )  e.  dom  R1
12 r1ord3g 7447 . . . . . 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 653 . . . . 5  |-  ( (
rank `  B )  C_  ( rank `  A
)  ->  ( R1 `  ( rank `  B
) )  C_  ( R1 `  ( rank `  A
) ) )
14 r1rankidb 7472 . . . . . . 7  |-  ( B  e.  U. ( R1
" On )  ->  B  C_  ( R1 `  ( rank `  B )
) )
1514sselda 3181 . . . . . 6  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  A  e.  ( R1
`  ( rank `  B
) ) )
16 ssel 3175 . . . . . 6  |-  ( ( R1 `  ( rank `  B ) )  C_  ( R1 `  ( rank `  A ) )  -> 
( A  e.  ( R1 `  ( rank `  B ) )  ->  A  e.  ( R1 `  ( rank `  A
) ) ) )
1715, 16syl5com 26 . . . . 5  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  ( ( R1 `  ( rank `  B )
)  C_  ( R1 `  ( rank `  A
) )  ->  A  e.  ( R1 `  ( rank `  A ) ) ) )
1813, 17syl5 28 . . . 4  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  ( ( rank `  B
)  C_  ( rank `  A )  ->  A  e.  ( R1 `  ( rank `  A ) ) ) )
199, 18syl5bir 209 . . 3  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  ( -.  ( rank `  A )  e.  (
rank `  B )  ->  A  e.  ( R1
`  ( rank `  A
) ) ) )
205, 19mt3d 117 . 2  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  ( rank `  A
)  e.  ( rank `  B ) )
2120ex 423 1  |-  ( B  e.  U. ( R1
" On )  -> 
( A  e.  B  ->  ( rank `  A
)  e.  ( rank `  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1685    C_ wss 3153   U.cuni 3828   Oncon0 4391    dom cdm 4688   "cima 4691   ` cfv 5221   R1cr1 7430   rankcrnk 7431
This theorem is referenced by:  wfelirr  7493  rankval3b  7494  rankel  7507  rankunb  7518  rankuni2b  7521  rankcf  8395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-recs 6384  df-rdg 6419  df-r1 7432  df-rank 7433
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