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Theorem rankeq0b 8191
Description: A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankeq0b  |-  ( A  e.  U. ( R1
" On )  -> 
( A  =  (/)  <->  ( rank `  A )  =  (/) ) )

Proof of Theorem rankeq0b
StepHypRef Expression
1 fveq2 5774 . . 3  |-  ( A  =  (/)  ->  ( rank `  A )  =  (
rank `  (/) ) )
2 r1funlim 8097 . . . . . . 7  |-  ( Fun 
R1  /\  Lim  dom  R1 )
32simpri 460 . . . . . 6  |-  Lim  dom  R1
4 limomss 6604 . . . . . 6  |-  ( Lim 
dom  R1  ->  om  C_  dom  R1 )
53, 4ax-mp 5 . . . . 5  |-  om  C_  dom  R1
6 peano1 6618 . . . . 5  |-  (/)  e.  om
75, 6sselii 3414 . . . 4  |-  (/)  e.  dom  R1
8 rankonid 8160 . . . 4  |-  ( (/)  e.  dom  R1  <->  ( rank `  (/) )  =  (/) )
97, 8mpbi 208 . . 3  |-  ( rank `  (/) )  =  (/)
101, 9syl6eq 2439 . 2  |-  ( A  =  (/)  ->  ( rank `  A )  =  (/) )
11 eqimss 3469 . . . . . . 7  |-  ( (
rank `  A )  =  (/)  ->  ( rank `  A )  C_  (/) )
1211adantl 464 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  (/) )  ->  ( rank `  A )  C_  (/) )
13 simpl 455 . . . . . . 7  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  (/) )  ->  A  e.  U. ( R1 " On ) )
14 rankr1bg 8134 . . . . . . 7  |-  ( ( A  e.  U. ( R1 " On )  /\  (/) 
e.  dom  R1 )  ->  ( A  C_  ( R1 `  (/) )  <->  ( rank `  A )  C_  (/) ) )
1513, 7, 14sylancl 660 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  (/) )  ->  ( A  C_  ( R1 `  (/) )  <->  ( rank `  A
)  C_  (/) ) )
1612, 15mpbird 232 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  (/) )  ->  A  C_  ( R1 `  (/) ) )
17 r10 8099 . . . . 5  |-  ( R1
`  (/) )  =  (/)
1816, 17syl6sseq 3463 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  (/) )  ->  A  C_  (/) )
19 ss0 3743 . . . 4  |-  ( A 
C_  (/)  ->  A  =  (/) )
2018, 19syl 16 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  (/) )  ->  A  =  (/) )
2120ex 432 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( ( rank `  A
)  =  (/)  ->  A  =  (/) ) )
2210, 21impbid2 204 1  |-  ( A  e.  U. ( R1
" On )  -> 
( A  =  (/)  <->  ( rank `  A )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826    C_ wss 3389   (/)c0 3711   U.cuni 4163   Oncon0 4792   Lim wlim 4793   dom cdm 4913   "cima 4916   Fun wfun 5490   ` cfv 5496   omcom 6599   R1cr1 8093   rankcrnk 8094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-om 6600  df-recs 6960  df-rdg 6994  df-r1 8095  df-rank 8096
This theorem is referenced by:  rankeq0  8192
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