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Theorem rankeq1o 25360
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 6581 . . . . . . 7  |-  1o  =/=  (/)
2 neeq1 2529 . . . . . . 7  |-  ( (
rank `  A )  =  1o  ->  ( (
rank `  A )  =/=  (/)  <->  1o  =/=  (/) ) )
31, 2mpbiri 224 . . . . . 6  |-  ( (
rank `  A )  =  1o  ->  ( rank `  A )  =/=  (/) )
4 df-ne 2523 . . . . . 6  |-  ( (
rank `  A )  =/=  (/)  <->  -.  ( rank `  A )  =  (/) )
53, 4sylib 188 . . . . 5  |-  ( (
rank `  A )  =  1o  ->  -.  ( rank `  A )  =  (/) )
6 fvprc 5602 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
75, 6nsyl2 119 . . . 4  |-  ( (
rank `  A )  =  1o  ->  A  e. 
_V )
8 fveq2 5608 . . . . . . 7  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
98eqeq1d 2366 . . . . . 6  |-  ( x  =  A  ->  (
( rank `  x )  =  1o  <->  ( rank `  A
)  =  1o ) )
10 eqeq1 2364 . . . . . 6  |-  ( x  =  A  ->  (
x  =  1o  <->  A  =  1o ) )
119, 10imbi12d 311 . . . . 5  |-  ( x  =  A  ->  (
( ( rank `  x
)  =  1o  ->  x  =  1o )  <->  ( ( rank `  A )  =  1o  ->  A  =  1o ) ) )
12 neeq1 2529 . . . . . . . 8  |-  ( (
rank `  x )  =  1o  ->  ( (
rank `  x )  =/=  (/)  <->  1o  =/=  (/) ) )
131, 12mpbiri 224 . . . . . . 7  |-  ( (
rank `  x )  =  1o  ->  ( rank `  x )  =/=  (/) )
14 vex 2867 . . . . . . . . 9  |-  x  e. 
_V
1514rankeq0 7623 . . . . . . . 8  |-  ( x  =  (/)  <->  ( rank `  x
)  =  (/) )
1615necon3bii 2553 . . . . . . 7  |-  ( x  =/=  (/)  <->  ( rank `  x
)  =/=  (/) )
1713, 16sylibr 203 . . . . . 6  |-  ( (
rank `  x )  =  1o  ->  x  =/=  (/) )
1814rankval 7578 . . . . . . . 8  |-  ( rank `  x )  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }
1918eqeq1i 2365 . . . . . . 7  |-  ( (
rank `  x )  =  1o  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  1o )
20 ssrab2 3334 . . . . . . . . . . 11  |-  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  C_  On
21 1on 6573 . . . . . . . . . . . . . . 15  |-  1o  e.  On
2221onirri 4581 . . . . . . . . . . . . . 14  |-  -.  1o  e.  1o
23 df1o2 6578 . . . . . . . . . . . . . . . 16  |-  1o  =  { (/) }
24 p0ex 4278 . . . . . . . . . . . . . . . 16  |-  { (/) }  e.  _V
2523, 24eqeltri 2428 . . . . . . . . . . . . . . 15  |-  1o  e.  _V
26 id 19 . . . . . . . . . . . . . . 15  |-  ( _V  =  1o  ->  _V  =  1o )
2725, 26syl5eleq 2444 . . . . . . . . . . . . . 14  |-  ( _V  =  1o  ->  1o  e.  1o )
2822, 27mto 167 . . . . . . . . . . . . 13  |-  -.  _V  =  1o
29 inteq 3946 . . . . . . . . . . . . . . 15  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  |^| (/) )
30 int0 3957 . . . . . . . . . . . . . . 15  |-  |^| (/)  =  _V
3129, 30syl6eq 2406 . . . . . . . . . . . . . 14  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  _V )
3231eqeq1d 2366 . . . . . . . . . . . . 13  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  <->  _V  =  1o ) )
3328, 32mtbiri 294 . . . . . . . . . . . 12  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  -.  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o )
3433necon2ai 2566 . . . . . . . . . . 11  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =/=  (/) )
35 onint 4668 . . . . . . . . . . 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 ) } )
3620, 34, 35sylancr 644 . . . . . . . . . 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 ) } )
37 eleq1 2418 . . . . . . . . . 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 ) } ) )
3836, 37mpbid 201 . . . . . . . . 9  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  1o  e.  { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
39 suceq 4539 . . . . . . . . . . . . 13  |-  ( y  =  1o  ->  suc  y  =  suc  1o )
4039fveq2d 5612 . . . . . . . . . . . 12  |-  ( y  =  1o  ->  ( R1 `  suc  y )  =  ( R1 `  suc  1o ) )
41 df-1o 6566 . . . . . . . . . . . . . . . . 17  |-  1o  =  suc  (/)
4241fveq2i 5611 . . . . . . . . . . . . . . . 16  |-  ( R1
`  1o )  =  ( R1 `  suc  (/) )
43 0elon 4527 . . . . . . . . . . . . . . . . 17  |-  (/)  e.  On
44 r1suc 7532 . . . . . . . . . . . . . . . . 17  |-  ( (/)  e.  On  ->  ( R1 ` 
suc  (/) )  =  ~P ( R1 `  (/) ) )
4543, 44ax-mp 8 . . . . . . . . . . . . . . . 16  |-  ( R1
`  suc  (/) )  =  ~P ( R1 `  (/) )
46 r10 7530 . . . . . . . . . . . . . . . . 17  |-  ( R1
`  (/) )  =  (/)
4746pweqi 3705 . . . . . . . . . . . . . . . 16  |-  ~P ( R1 `  (/) )  =  ~P (/)
4842, 45, 473eqtri 2382 . . . . . . . . . . . . . . 15  |-  ( R1
`  1o )  =  ~P (/)
4948pweqi 3705 . . . . . . . . . . . . . 14  |-  ~P ( R1 `  1o )  =  ~P ~P (/)
50 pw0 3841 . . . . . . . . . . . . . . 15  |-  ~P (/)  =  { (/)
}
5150pweqi 3705 . . . . . . . . . . . . . 14  |-  ~P ~P (/)  =  ~P { (/) }
52 pwpw0 3842 . . . . . . . . . . . . . 14  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
5349, 51, 523eqtrri 2383 . . . . . . . . . . . . 13  |-  { (/) ,  { (/) } }  =  ~P ( R1 `  1o )
54 r1suc 7532 . . . . . . . . . . . . . 14  |-  ( 1o  e.  On  ->  ( R1 `  suc  1o )  =  ~P ( R1
`  1o ) )
5521, 54ax-mp 8 . . . . . . . . . . . . 13  |-  ( R1
`  suc  1o )  =  ~P ( R1 `  1o )
5653, 55eqtr4i 2381 . . . . . . . . . . . 12  |-  { (/) ,  { (/) } }  =  ( R1 `  suc  1o )
5740, 56syl6eqr 2408 . . . . . . . . . . 11  |-  ( y  =  1o  ->  ( R1 `  suc  y )  =  { (/) ,  { (/)
} } )
5857eleq2d 2425 . . . . . . . . . 10  |-  ( y  =  1o  ->  (
x  e.  ( R1
`  suc  y )  <->  x  e.  { (/) ,  { (/)
} } ) )
5958elrab 2999 . . . . . . . . 9  |-  ( 1o  e.  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  <->  ( 1o  e.  On  /\  x  e. 
{ (/) ,  { (/) } } ) )
6038, 59sylib 188 . . . . . . . 8  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  ( 1o  e.  On  /\  x  e.  { (/) ,  { (/) } } ) )
6114elpr 3734 . . . . . . . . . 10  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
62 df-ne 2523 . . . . . . . . . . . 12  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
63 orel1 371 . . . . . . . . . . . 12  |-  ( -.  x  =  (/)  ->  (
( x  =  (/)  \/  x  =  { (/) } )  ->  x  =  { (/) } ) )
6462, 63sylbi 187 . . . . . . . . . . 11  |-  ( x  =/=  (/)  ->  ( (
x  =  (/)  \/  x  =  { (/) } )  ->  x  =  { (/) } ) )
65 eqeq2 2367 . . . . . . . . . . . . 13  |-  ( x  =  { (/) }  ->  ( 1o  =  x  <->  1o  =  { (/) } ) )
6623, 65mpbiri 224 . . . . . . . . . . . 12  |-  ( x  =  { (/) }  ->  1o  =  x )
6766eqcomd 2363 . . . . . . . . . . 11  |-  ( x  =  { (/) }  ->  x  =  1o )
6864, 67syl6com 31 . . . . . . . . . 10  |-  ( ( x  =  (/)  \/  x  =  { (/) } )  -> 
( x  =/=  (/)  ->  x  =  1o ) )
6961, 68sylbi 187 . . . . . . . . 9  |-  ( x  e.  { (/) ,  { (/)
} }  ->  (
x  =/=  (/)  ->  x  =  1o ) )
7069adantl 452 . . . . . . . 8  |-  ( ( 1o  e.  On  /\  x  e.  { (/) ,  { (/)
} } )  -> 
( x  =/=  (/)  ->  x  =  1o ) )
7160, 70syl 15 . . . . . . 7  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  (
x  =/=  (/)  ->  x  =  1o ) )
7219, 71sylbi 187 . . . . . 6  |-  ( (
rank `  x )  =  1o  ->  ( x  =/=  (/)  ->  x  =  1o ) )
7317, 72mpd 14 . . . . 5  |-  ( (
rank `  x )  =  1o  ->  x  =  1o )
7411, 73vtoclg 2919 . . . 4  |-  ( A  e.  _V  ->  (
( rank `  A )  =  1o  ->  A  =  1o ) )
757, 74mpcom 32 . . 3  |-  ( (
rank `  A )  =  1o  ->  A  =  1o )
76 fveq2 5608 . . . 4  |-  ( A  =  1o  ->  ( rank `  A )  =  ( rank `  1o ) )
77 r111 7537 . . . . . . 7  |-  R1 : On
-1-1-> _V
78 f1dm 5524 . . . . . . 7  |-  ( R1 : On -1-1-> _V  ->  dom 
R1  =  On )
7977, 78ax-mp 8 . . . . . 6  |-  dom  R1  =  On
8021, 79eleqtrri 2431 . . . . 5  |-  1o  e.  dom  R1
81 rankonid 7591 . . . . 5  |-  ( 1o  e.  dom  R1  <->  ( rank `  1o )  =  1o )
8280, 81mpbi 199 . . . 4  |-  ( rank `  1o )  =  1o
8376, 82syl6eq 2406 . . 3  |-  ( A  =  1o  ->  ( rank `  A )  =  1o )
8475, 83impbii 180 . 2  |-  ( (
rank `  A )  =  1o  <->  A  =  1o )
8523eqeq2i 2368 . 2  |-  ( A  =  1o  <->  A  =  { (/) } )
8684, 85bitri 240 1  |-  ( (
rank `  A )  =  1o  <->  A  =  { (/)
} )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   {crab 2623   _Vcvv 2864    C_ wss 3228   (/)c0 3531   ~Pcpw 3701   {csn 3716   {cpr 3717   |^|cint 3943   Oncon0 4474   suc csuc 4476   dom cdm 4771   -1-1->wf1 5334   ` cfv 5337   1oc1o 6559   R1cr1 7524   rankcrnk 7525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-reg 7396  ax-inf2 7432
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-recs 6475  df-rdg 6510  df-1o 6566  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-r1 7526  df-rank 7527
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