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Theorem rankelb 7380
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 7361 . . . . . 6  |-  ( B  e.  U. ( R1
" On )  ->  B  C_  U. ( R1
" On ) )
21sseld 3102 . . . . 5  |-  ( B  e.  U. ( R1
" On )  -> 
( A  e.  B  ->  A  e.  U. ( R1 " On ) ) )
3 rankidn 7378 . . . . 5  |-  ( A  e.  U. ( R1
" On )  ->  -.  A  e.  ( R1 `  ( rank `  A
) ) )
42, 3syl6 31 . . . 4  |-  ( B  e.  U. ( R1
" On )  -> 
( A  e.  B  ->  -.  A  e.  ( R1 `  ( rank `  A ) ) ) )
54imp 420 . . 3  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  -.  A  e.  ( R1 `  ( rank `  A ) ) )
6 rankon 7351 . . . . 5  |-  ( rank `  B )  e.  On
7 rankon 7351 . . . . 5  |-  ( rank `  A )  e.  On
8 ontri1 4319 . . . . 5  |-  ( ( ( rank `  B
)  e.  On  /\  ( rank `  A )  e.  On )  ->  (
( rank `  B )  C_  ( rank `  A
)  <->  -.  ( rank `  A )  e.  (
rank `  B )
) )
96, 7, 8mp2an 656 . . . 4  |-  ( (
rank `  B )  C_  ( rank `  A
)  <->  -.  ( rank `  A )  e.  (
rank `  B )
)
10 rankdmr1 7357 . . . . . 6  |-  ( rank `  B )  e.  dom  R1
11 rankdmr1 7357 . . . . . 6  |-  ( rank `  A )  e.  dom  R1
12 r1ord3g 7335 . . . . . 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 656 . . . . 5  |-  ( (
rank `  B )  C_  ( rank `  A
)  ->  ( R1 `  ( rank `  B
) )  C_  ( R1 `  ( rank `  A
) ) )
14 r1rankidb 7360 . . . . . . 7  |-  ( B  e.  U. ( R1
" On )  ->  B  C_  ( R1 `  ( rank `  B )
) )
1514sselda 3103 . . . . . 6  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  A  e.  ( R1
`  ( rank `  B
) ) )
16 ssel 3097 . . . . . 6  |-  ( ( R1 `  ( rank `  B ) )  C_  ( R1 `  ( rank `  A ) )  -> 
( A  e.  ( R1 `  ( rank `  B ) )  ->  A  e.  ( R1 `  ( rank `  A
) ) ) )
1715, 16syl5com 28 . . . . 5  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  ( ( R1 `  ( rank `  B )
)  C_  ( R1 `  ( rank `  A
) )  ->  A  e.  ( R1 `  ( rank `  A ) ) ) )
1813, 17syl5 30 . . . 4  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  ( ( rank `  B
)  C_  ( rank `  A )  ->  A  e.  ( R1 `  ( rank `  A ) ) ) )
199, 18syl5bir 211 . . 3  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  ( -.  ( rank `  A )  e.  (
rank `  B )  ->  A  e.  ( R1
`  ( rank `  A
) ) ) )
205, 19mt3d 119 . 2  |-  ( ( B  e.  U. ( R1 " On )  /\  A  e.  B )  ->  ( rank `  A
)  e.  ( rank `  B ) )
2120ex 425 1  |-  ( B  e.  U. ( R1
" On )  -> 
( A  e.  B  ->  ( rank `  A
)  e.  ( rank `  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1621    C_ wss 3078   U.cuni 3727   Oncon0 4285   dom cdm 4580   "cima 4583   ` cfv 4592   R1cr1 7318   rankcrnk 7319
This theorem is referenced by:  wfelirr  7381  rankval3b  7382  rankel  7395  rankunb  7406  rankuni2b  7409  rankcf  8279
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-recs 6274  df-rdg 6309  df-r1 7320  df-rank 7321
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