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Theorem rankelb 7498
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 7479 . . . . . 6  |-  ( B  e.  U. ( R1
" On )  ->  B  C_  U. ( R1
" On ) )
21sseld 3181 . . . . 5  |-  ( B  e.  U. ( R1
" On )  -> 
( A  e.  B  ->  A  e.  U. ( R1 " On ) ) )
3 rankidn 7496 . . . . 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 7469 . . . . 5  |-  ( rank `  B )  e.  On
7 rankon 7469 . . . . 5  |-  ( rank `  A )  e.  On
8 ontri1 4428 . . . . 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 7475 . . . . . 6  |-  ( rank `  B )  e.  dom  R1
11 rankdmr1 7475 . . . . . 6  |-  ( rank `  A )  e.  dom  R1
12 r1ord3g 7453 . . . . . 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 7478 . . . . . . 7  |-  ( B  e.  U. ( R1
" On )  ->  B  C_  ( R1 `  ( rank `  B )
) )
1514sselda 3182 . . . . . 6  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  A  e.  ( R1
`  ( rank `  B
) ) )
16 ssel 3176 . . . . . 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 1686    C_ wss 3154   U.cuni 3829   Oncon0 4394   dom cdm 4691   "cima 4694   ` cfv 5257   R1cr1 7436   rankcrnk 7437
This theorem is referenced by:  wfelirr  7499  rankval3b  7500  rankel  7513  rankunb  7524  rankuni2b  7527  rankcf  8401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-recs 6390  df-rdg 6425  df-r1 7438  df-rank 7439
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