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Theorem rankssb 7404
Description: The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankssb  |-  ( B  e.  U. ( R1
" On )  -> 
( A  C_  B  ->  ( rank `  A
)  C_  ( rank `  B ) ) )

Proof of Theorem rankssb
StepHypRef Expression
1 simpr 449 . . . 4  |-  ( ( B  e.  U. ( R1 " On )  /\  A  C_  B )  ->  A  C_  B )
2 r1rankidb 7360 . . . . 5  |-  ( B  e.  U. ( R1
" On )  ->  B  C_  ( R1 `  ( rank `  B )
) )
32adantr 453 . . . 4  |-  ( ( B  e.  U. ( R1 " On )  /\  A  C_  B )  ->  B  C_  ( R1 `  ( rank `  B )
) )
41, 3sstrd 3110 . . 3  |-  ( ( B  e.  U. ( R1 " On )  /\  A  C_  B )  ->  A  C_  ( R1 `  ( rank `  B )
) )
5 sswf 7364 . . . 4  |-  ( ( B  e.  U. ( R1 " On )  /\  A  C_  B )  ->  A  e.  U. ( R1 " On ) )
6 rankdmr1 7357 . . . 4  |-  ( rank `  B )  e.  dom  R1
7 rankr1bg 7359 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  B )  e.  dom  R1 )  -> 
( A  C_  ( R1 `  ( rank `  B
) )  <->  ( rank `  A )  C_  ( rank `  B ) ) )
85, 6, 7sylancl 646 . . 3  |-  ( ( B  e.  U. ( R1 " On )  /\  A  C_  B )  -> 
( A  C_  ( R1 `  ( rank `  B
) )  <->  ( rank `  A )  C_  ( rank `  B ) ) )
94, 8mpbid 203 . 2  |-  ( ( B  e.  U. ( R1 " On )  /\  A  C_  B )  -> 
( rank `  A )  C_  ( rank `  B
) )
109ex 425 1  |-  ( B  e.  U. ( R1
" On )  -> 
( A  C_  B  ->  ( rank `  A
)  C_  ( rank `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> 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:  rankss  7405  rankunb  7406  rankuni2b  7409  rankr1id  7418
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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