MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rankssb Unicode version

Theorem rankssb 7520
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 447 . . . 4  |-  ( ( B  e.  U. ( R1 " On )  /\  A  C_  B )  ->  A  C_  B )
2 r1rankidb 7476 . . . . 5  |-  ( B  e.  U. ( R1
" On )  ->  B  C_  ( R1 `  ( rank `  B )
) )
32adantr 451 . . . 4  |-  ( ( B  e.  U. ( R1 " On )  /\  A  C_  B )  ->  B  C_  ( R1 `  ( rank `  B )
) )
41, 3sstrd 3189 . . 3  |-  ( ( B  e.  U. ( R1 " On )  /\  A  C_  B )  ->  A  C_  ( R1 `  ( rank `  B )
) )
5 sswf 7480 . . . 4  |-  ( ( B  e.  U. ( R1 " On )  /\  A  C_  B )  ->  A  e.  U. ( R1 " On ) )
6 rankdmr1 7473 . . . 4  |-  ( rank `  B )  e.  dom  R1
7 rankr1bg 7475 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  B )  e.  dom  R1 )  -> 
( A  C_  ( R1 `  ( rank `  B
) )  <->  ( rank `  A )  C_  ( rank `  B ) ) )
85, 6, 7sylancl 643 . . 3  |-  ( ( B  e.  U. ( R1 " On )  /\  A  C_  B )  -> 
( A  C_  ( R1 `  ( rank `  B
) )  <->  ( rank `  A )  C_  ( rank `  B ) ) )
94, 8mpbid 201 . 2  |-  ( ( B  e.  U. ( R1 " On )  /\  A  C_  B )  -> 
( rank `  A )  C_  ( rank `  B
) )
109ex 423 1  |-  ( B  e.  U. ( R1
" On )  -> 
( A  C_  B  ->  ( rank `  A
)  C_  ( rank `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    C_ wss 3152   U.cuni 3827   Oncon0 4392   dom cdm 4689   "cima 4692   ` cfv 5255   R1cr1 7434   rankcrnk 7435
This theorem is referenced by:  rankss  7521  rankunb  7522  rankuni2b  7525  rankr1id  7534
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437
  Copyright terms: Public domain W3C validator