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Theorem rankeq0b 7788
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 5730 . . 3  |-  ( A  =  (/)  ->  ( rank `  A )  =  (
rank `  (/) ) )
2 r1funlim 7694 . . . . . . 7  |-  ( Fun 
R1  /\  Lim  dom  R1 )
32simpri 450 . . . . . 6  |-  Lim  dom  R1
4 limomss 4852 . . . . . 6  |-  ( Lim 
dom  R1  ->  om  C_  dom  R1 )
53, 4ax-mp 8 . . . . 5  |-  om  C_  dom  R1
6 peano1 4866 . . . . 5  |-  (/)  e.  om
75, 6sselii 3347 . . . 4  |-  (/)  e.  dom  R1
8 rankonid 7757 . . . 4  |-  ( (/)  e.  dom  R1  <->  ( rank `  (/) )  =  (/) )
97, 8mpbi 201 . . 3  |-  ( rank `  (/) )  =  (/)
101, 9syl6eq 2486 . 2  |-  ( A  =  (/)  ->  ( rank `  A )  =  (/) )
11 eqimss 3402 . . . . . . 7  |-  ( (
rank `  A )  =  (/)  ->  ( rank `  A )  C_  (/) )
1211adantl 454 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  (/) )  ->  ( rank `  A )  C_  (/) )
13 simpl 445 . . . . . . 7  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  (/) )  ->  A  e.  U. ( R1 " On ) )
14 rankr1bg 7731 . . . . . . 7  |-  ( ( A  e.  U. ( R1 " On )  /\  (/) 
e.  dom  R1 )  ->  ( A  C_  ( R1 `  (/) )  <->  ( rank `  A )  C_  (/) ) )
1513, 7, 14sylancl 645 . . . . . 6  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  (/) )  ->  ( A  C_  ( R1 `  (/) )  <->  ( rank `  A
)  C_  (/) ) )
1612, 15mpbird 225 . . . . 5  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  (/) )  ->  A  C_  ( R1 `  (/) ) )
17 r10 7696 . . . . 5  |-  ( R1
`  (/) )  =  (/)
1816, 17syl6sseq 3396 . . . 4  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  (/) )  ->  A  C_  (/) )
19 ss0 3660 . . . 4  |-  ( A 
C_  (/)  ->  A  =  (/) )
2018, 19syl 16 . . 3  |-  ( ( A  e.  U. ( R1 " On )  /\  ( rank `  A )  =  (/) )  ->  A  =  (/) )
2120ex 425 . 2  |-  ( A  e.  U. ( R1
" On )  -> 
( ( rank `  A
)  =  (/)  ->  A  =  (/) ) )
2210, 21impbid2 197 1  |-  ( A  e.  U. ( R1
" On )  -> 
( A  =  (/)  <->  ( rank `  A )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   (/)c0 3630   U.cuni 4017   Oncon0 4583   Lim wlim 4584   omcom 4847   dom cdm 4880   "cima 4883   Fun wfun 5450   ` cfv 5456   R1cr1 7690   rankcrnk 7691
This theorem is referenced by:  rankeq0  7789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-recs 6635  df-rdg 6670  df-r1 7692  df-rank 7693
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