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Theorem rankeq1o 28207
Description: The only set with rank  1o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)
Assertion
Ref Expression
rankeq1o  |-  ( (
rank `  A )  =  1o  <->  A  =  { (/)
} )

Proof of Theorem rankeq1o
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 6933 . . . . . . 7  |-  1o  =/=  (/)
2 neeq1 2614 . . . . . . 7  |-  ( (
rank `  A )  =  1o  ->  ( (
rank `  A )  =/=  (/)  <->  1o  =/=  (/) ) )
31, 2mpbiri 233 . . . . . 6  |-  ( (
rank `  A )  =  1o  ->  ( rank `  A )  =/=  (/) )
43neneqd 2622 . . . . 5  |-  ( (
rank `  A )  =  1o  ->  -.  ( rank `  A )  =  (/) )
5 fvprc 5683 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
64, 5nsyl2 127 . . . 4  |-  ( (
rank `  A )  =  1o  ->  A  e. 
_V )
7 fveq2 5689 . . . . . . 7  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
87eqeq1d 2449 . . . . . 6  |-  ( x  =  A  ->  (
( rank `  x )  =  1o  <->  ( rank `  A
)  =  1o ) )
9 eqeq1 2447 . . . . . 6  |-  ( x  =  A  ->  (
x  =  1o  <->  A  =  1o ) )
108, 9imbi12d 320 . . . . 5  |-  ( x  =  A  ->  (
( ( rank `  x
)  =  1o  ->  x  =  1o )  <->  ( ( rank `  A )  =  1o  ->  A  =  1o ) ) )
11 neeq1 2614 . . . . . . . 8  |-  ( (
rank `  x )  =  1o  ->  ( (
rank `  x )  =/=  (/)  <->  1o  =/=  (/) ) )
121, 11mpbiri 233 . . . . . . 7  |-  ( (
rank `  x )  =  1o  ->  ( rank `  x )  =/=  (/) )
13 vex 2973 . . . . . . . . 9  |-  x  e. 
_V
1413rankeq0 8066 . . . . . . . 8  |-  ( x  =  (/)  <->  ( rank `  x
)  =  (/) )
1514necon3bii 2638 . . . . . . 7  |-  ( x  =/=  (/)  <->  ( rank `  x
)  =/=  (/) )
1612, 15sylibr 212 . . . . . 6  |-  ( (
rank `  x )  =  1o  ->  x  =/=  (/) )
1713rankval 8021 . . . . . . . 8  |-  ( rank `  x )  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }
1817eqeq1i 2448 . . . . . . 7  |-  ( (
rank `  x )  =  1o  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  1o )
19 ssrab2 3435 . . . . . . . . . . 11  |-  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  C_  On
20 elirr 7811 . . . . . . . . . . . . . 14  |-  -.  1o  e.  1o
21 df1o2 6930 . . . . . . . . . . . . . . . 16  |-  1o  =  { (/) }
22 p0ex 4477 . . . . . . . . . . . . . . . 16  |-  { (/) }  e.  _V
2321, 22eqeltri 2511 . . . . . . . . . . . . . . 15  |-  1o  e.  _V
24 id 22 . . . . . . . . . . . . . . 15  |-  ( _V  =  1o  ->  _V  =  1o )
2523, 24syl5eleq 2527 . . . . . . . . . . . . . 14  |-  ( _V  =  1o  ->  1o  e.  1o )
2620, 25mto 176 . . . . . . . . . . . . 13  |-  -.  _V  =  1o
27 inteq 4129 . . . . . . . . . . . . . . 15  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  |^| (/) )
28 int0 4140 . . . . . . . . . . . . . . 15  |-  |^| (/)  =  _V
2927, 28syl6eq 2489 . . . . . . . . . . . . . 14  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  _V )
3029eqeq1d 2449 . . . . . . . . . . . . 13  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  <->  _V  =  1o ) )
3126, 30mtbiri 303 . . . . . . . . . . . 12  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  -.  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o )
3231necon2ai 2654 . . . . . . . . . . 11  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =/=  (/) )
33 onint 6404 . . . . . . . . . . 11  |-  ( ( { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  C_  On  /\  {
y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =/=  (/) )  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } )
3419, 32, 33sylancr 663 . . . . . . . . . 10  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } )
35 eleq1 2501 . . . . . . . . . 10  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  ( |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  e.  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  <->  1o  e.  { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) } ) )
3634, 35mpbid 210 . . . . . . . . 9  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  1o  e.  { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
37 suceq 4782 . . . . . . . . . . . . 13  |-  ( y  =  1o  ->  suc  y  =  suc  1o )
3837fveq2d 5693 . . . . . . . . . . . 12  |-  ( y  =  1o  ->  ( R1 `  suc  y )  =  ( R1 `  suc  1o ) )
39 df-1o 6918 . . . . . . . . . . . . . . . . 17  |-  1o  =  suc  (/)
4039fveq2i 5692 . . . . . . . . . . . . . . . 16  |-  ( R1
`  1o )  =  ( R1 `  suc  (/) )
41 0elon 4770 . . . . . . . . . . . . . . . . 17  |-  (/)  e.  On
42 r1suc 7975 . . . . . . . . . . . . . . . . 17  |-  ( (/)  e.  On  ->  ( R1 ` 
suc  (/) )  =  ~P ( R1 `  (/) ) )
4341, 42ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( R1
`  suc  (/) )  =  ~P ( R1 `  (/) )
44 r10 7973 . . . . . . . . . . . . . . . . 17  |-  ( R1
`  (/) )  =  (/)
4544pweqi 3862 . . . . . . . . . . . . . . . 16  |-  ~P ( R1 `  (/) )  =  ~P (/)
4640, 43, 453eqtri 2465 . . . . . . . . . . . . . . 15  |-  ( R1
`  1o )  =  ~P (/)
4746pweqi 3862 . . . . . . . . . . . . . 14  |-  ~P ( R1 `  1o )  =  ~P ~P (/)
48 pw0 4018 . . . . . . . . . . . . . . 15  |-  ~P (/)  =  { (/)
}
4948pweqi 3862 . . . . . . . . . . . . . 14  |-  ~P ~P (/)  =  ~P { (/) }
50 pwpw0 4019 . . . . . . . . . . . . . 14  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
5147, 49, 503eqtrri 2466 . . . . . . . . . . . . 13  |-  { (/) ,  { (/) } }  =  ~P ( R1 `  1o )
52 1on 6925 . . . . . . . . . . . . . 14  |-  1o  e.  On
53 r1suc 7975 . . . . . . . . . . . . . 14  |-  ( 1o  e.  On  ->  ( R1 `  suc  1o )  =  ~P ( R1
`  1o ) )
5452, 53ax-mp 5 . . . . . . . . . . . . 13  |-  ( R1
`  suc  1o )  =  ~P ( R1 `  1o )
5551, 54eqtr4i 2464 . . . . . . . . . . . 12  |-  { (/) ,  { (/) } }  =  ( R1 `  suc  1o )
5638, 55syl6eqr 2491 . . . . . . . . . . 11  |-  ( y  =  1o  ->  ( R1 `  suc  y )  =  { (/) ,  { (/)
} } )
5756eleq2d 2508 . . . . . . . . . 10  |-  ( y  =  1o  ->  (
x  e.  ( R1
`  suc  y )  <->  x  e.  { (/) ,  { (/)
} } ) )
5857elrab 3115 . . . . . . . . 9  |-  ( 1o  e.  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  <->  ( 1o  e.  On  /\  x  e. 
{ (/) ,  { (/) } } ) )
5936, 58sylib 196 . . . . . . . 8  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  ( 1o  e.  On  /\  x  e.  { (/) ,  { (/) } } ) )
6013elpr 3893 . . . . . . . . . 10  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
61 df-ne 2606 . . . . . . . . . . . 12  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
62 orel1 382 . . . . . . . . . . . 12  |-  ( -.  x  =  (/)  ->  (
( x  =  (/)  \/  x  =  { (/) } )  ->  x  =  { (/) } ) )
6361, 62sylbi 195 . . . . . . . . . . 11  |-  ( x  =/=  (/)  ->  ( (
x  =  (/)  \/  x  =  { (/) } )  ->  x  =  { (/) } ) )
64 eqeq2 2450 . . . . . . . . . . . . 13  |-  ( x  =  { (/) }  ->  ( 1o  =  x  <->  1o  =  { (/) } ) )
6521, 64mpbiri 233 . . . . . . . . . . . 12  |-  ( x  =  { (/) }  ->  1o  =  x )
6665eqcomd 2446 . . . . . . . . . . 11  |-  ( x  =  { (/) }  ->  x  =  1o )
6763, 66syl6com 35 . . . . . . . . . 10  |-  ( ( x  =  (/)  \/  x  =  { (/) } )  -> 
( x  =/=  (/)  ->  x  =  1o ) )
6860, 67sylbi 195 . . . . . . . . 9  |-  ( x  e.  { (/) ,  { (/)
} }  ->  (
x  =/=  (/)  ->  x  =  1o ) )
6968adantl 466 . . . . . . . 8  |-  ( ( 1o  e.  On  /\  x  e.  { (/) ,  { (/)
} } )  -> 
( x  =/=  (/)  ->  x  =  1o ) )
7059, 69syl 16 . . . . . . 7  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  (
x  =/=  (/)  ->  x  =  1o ) )
7118, 70sylbi 195 . . . . . 6  |-  ( (
rank `  x )  =  1o  ->  ( x  =/=  (/)  ->  x  =  1o ) )
7216, 71mpd 15 . . . . 5  |-  ( (
rank `  x )  =  1o  ->  x  =  1o )
7310, 72vtoclg 3028 . . . 4  |-  ( A  e.  _V  ->  (
( rank `  A )  =  1o  ->  A  =  1o ) )
746, 73mpcom 36 . . 3  |-  ( (
rank `  A )  =  1o  ->  A  =  1o )
75 fveq2 5689 . . . 4  |-  ( A  =  1o  ->  ( rank `  A )  =  ( rank `  1o ) )
76 r111 7980 . . . . . . 7  |-  R1 : On
-1-1-> _V
77 f1dm 5608 . . . . . . 7  |-  ( R1 : On -1-1-> _V  ->  dom 
R1  =  On )
7876, 77ax-mp 5 . . . . . 6  |-  dom  R1  =  On
7952, 78eleqtrri 2514 . . . . 5  |-  1o  e.  dom  R1
80 rankonid 8034 . . . . 5  |-  ( 1o  e.  dom  R1  <->  ( rank `  1o )  =  1o )
8179, 80mpbi 208 . . . 4  |-  ( rank `  1o )  =  1o
8275, 81syl6eq 2489 . . 3  |-  ( A  =  1o  ->  ( rank `  A )  =  1o )
8374, 82impbii 188 . 2  |-  ( (
rank `  A )  =  1o  <->  A  =  1o )
8421eqeq2i 2451 . 2  |-  ( A  =  1o  <->  A  =  { (/) } )
8583, 84bitri 249 1  |-  ( (
rank `  A )  =  1o  <->  A  =  { (/)
} )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   {crab 2717   _Vcvv 2970    C_ wss 3326   (/)c0 3635   ~Pcpw 3858   {csn 3875   {cpr 3877   |^|cint 4126   Oncon0 4717   suc csuc 4719   dom cdm 4838   -1-1->wf1 5413   ` cfv 5416   1oc1o 6911   R1cr1 7967   rankcrnk 7968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-reg 7805  ax-inf2 7845
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-om 6475  df-recs 6830  df-rdg 6864  df-1o 6918  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-r1 7969  df-rank 7970
This theorem is referenced by: (None)
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