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Theorem sswf 8217
Description: A subset of a well-founded set is well-founded. (Contributed by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
sswf  |-  ( ( A  e.  U. ( R1 " On )  /\  B  C_  A )  ->  B  e.  U. ( R1 " On ) )

Proof of Theorem sswf
StepHypRef Expression
1 rankidb 8209 . . 3  |-  ( A  e.  U. ( R1
" On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
2 r1sscl 8194 . . 3  |-  ( ( A  e.  ( R1
`  suc  ( rank `  A ) )  /\  B  C_  A )  ->  B  e.  ( R1 ` 
suc  ( rank `  A
) ) )
31, 2sylan 469 . 2  |-  ( ( A  e.  U. ( R1 " On )  /\  B  C_  A )  ->  B  e.  ( R1 ` 
suc  ( rank `  A
) ) )
4 r1elwf 8205 . 2  |-  ( B  e.  ( R1 `  suc  ( rank `  A
) )  ->  B  e.  U. ( R1 " On ) )
53, 4syl 16 1  |-  ( ( A  e.  U. ( R1 " On )  /\  B  C_  A )  ->  B  e.  U. ( R1 " On ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823    C_ wss 3461   U.cuni 4235   Oncon0 4867   suc csuc 4869   "cima 4991   ` cfv 5570   R1cr1 8171   rankcrnk 8172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173  df-rank 8174
This theorem is referenced by:  snwf  8218  unwf  8219  uniwf  8228  rankssb  8257
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