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Theorem rankeq1o 30723
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 7205 . . . . . . 7  |-  1o  =/=  (/)
2 neeq1 2712 . . . . . . 7  |-  ( (
rank `  A )  =  1o  ->  ( (
rank `  A )  =/=  (/)  <->  1o  =/=  (/) ) )
31, 2mpbiri 236 . . . . . 6  |-  ( (
rank `  A )  =  1o  ->  ( rank `  A )  =/=  (/) )
43neneqd 2632 . . . . 5  |-  ( (
rank `  A )  =  1o  ->  -.  ( rank `  A )  =  (/) )
5 fvprc 5875 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
64, 5nsyl2 130 . . . 4  |-  ( (
rank `  A )  =  1o  ->  A  e. 
_V )
7 fveq2 5881 . . . . . . 7  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
87eqeq1d 2431 . . . . . 6  |-  ( x  =  A  ->  (
( rank `  x )  =  1o  <->  ( rank `  A
)  =  1o ) )
9 eqeq1 2433 . . . . . 6  |-  ( x  =  A  ->  (
x  =  1o  <->  A  =  1o ) )
108, 9imbi12d 321 . . . . 5  |-  ( x  =  A  ->  (
( ( rank `  x
)  =  1o  ->  x  =  1o )  <->  ( ( rank `  A )  =  1o  ->  A  =  1o ) ) )
11 neeq1 2712 . . . . . . . 8  |-  ( (
rank `  x )  =  1o  ->  ( (
rank `  x )  =/=  (/)  <->  1o  =/=  (/) ) )
121, 11mpbiri 236 . . . . . . 7  |-  ( (
rank `  x )  =  1o  ->  ( rank `  x )  =/=  (/) )
13 vex 3090 . . . . . . . . 9  |-  x  e. 
_V
1413rankeq0 8331 . . . . . . . 8  |-  ( x  =  (/)  <->  ( rank `  x
)  =  (/) )
1514necon3bii 2699 . . . . . . 7  |-  ( x  =/=  (/)  <->  ( rank `  x
)  =/=  (/) )
1612, 15sylibr 215 . . . . . 6  |-  ( (
rank `  x )  =  1o  ->  x  =/=  (/) )
1713rankval 8286 . . . . . . . 8  |-  ( rank `  x )  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }
1817eqeq1i 2436 . . . . . . 7  |-  ( (
rank `  x )  =  1o  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  1o )
19 ssrab2 3552 . . . . . . . . . . 11  |-  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  C_  On
20 elirr 8113 . . . . . . . . . . . . . 14  |-  -.  1o  e.  1o
21 df1o2 7202 . . . . . . . . . . . . . . . 16  |-  1o  =  { (/) }
22 p0ex 4612 . . . . . . . . . . . . . . . 16  |-  { (/) }  e.  _V
2321, 22eqeltri 2513 . . . . . . . . . . . . . . 15  |-  1o  e.  _V
24 id 23 . . . . . . . . . . . . . . 15  |-  ( _V  =  1o  ->  _V  =  1o )
2523, 24syl5eleq 2523 . . . . . . . . . . . . . 14  |-  ( _V  =  1o  ->  1o  e.  1o )
2620, 25mto 179 . . . . . . . . . . . . 13  |-  -.  _V  =  1o
27 inteq 4261 . . . . . . . . . . . . . . 15  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  |^| (/) )
28 int0 4272 . . . . . . . . . . . . . . 15  |-  |^| (/)  =  _V
2927, 28syl6eq 2486 . . . . . . . . . . . . . 14  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  _V )
3029eqeq1d 2431 . . . . . . . . . . . . 13  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  <->  _V  =  1o ) )
3126, 30mtbiri 304 . . . . . . . . . . . 12  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  -.  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o )
3231necon2ai 2666 . . . . . . . . . . 11  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =/=  (/) )
33 onint 6636 . . . . . . . . . . 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 667 . . . . . . . . . 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 213 . . . . . . . . 9  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  1o  e.  { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
37 suceq 5507 . . . . . . . . . . . . 13  |-  ( y  =  1o  ->  suc  y  =  suc  1o )
3837fveq2d 5885 . . . . . . . . . . . 12  |-  ( y  =  1o  ->  ( R1 `  suc  y )  =  ( R1 `  suc  1o ) )
39 df-1o 7190 . . . . . . . . . . . . . . . . 17  |-  1o  =  suc  (/)
4039fveq2i 5884 . . . . . . . . . . . . . . . 16  |-  ( R1
`  1o )  =  ( R1 `  suc  (/) )
41 0elon 5495 . . . . . . . . . . . . . . . . 17  |-  (/)  e.  On
42 r1suc 8240 . . . . . . . . . . . . . . . . 17  |-  ( (/)  e.  On  ->  ( R1 ` 
suc  (/) )  =  ~P ( R1 `  (/) ) )
4341, 42ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( R1
`  suc  (/) )  =  ~P ( R1 `  (/) )
44 r10 8238 . . . . . . . . . . . . . . . . 17  |-  ( R1
`  (/) )  =  (/)
4544pweqi 3989 . . . . . . . . . . . . . . . 16  |-  ~P ( R1 `  (/) )  =  ~P (/)
4640, 43, 453eqtri 2462 . . . . . . . . . . . . . . 15  |-  ( R1
`  1o )  =  ~P (/)
4746pweqi 3989 . . . . . . . . . . . . . 14  |-  ~P ( R1 `  1o )  =  ~P ~P (/)
48 pw0 4150 . . . . . . . . . . . . . . 15  |-  ~P (/)  =  { (/)
}
4948pweqi 3989 . . . . . . . . . . . . . 14  |-  ~P ~P (/)  =  ~P { (/) }
50 pwpw0 4151 . . . . . . . . . . . . . 14  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
5147, 49, 503eqtrri 2463 . . . . . . . . . . . . 13  |-  { (/) ,  { (/) } }  =  ~P ( R1 `  1o )
52 1on 7197 . . . . . . . . . . . . . 14  |-  1o  e.  On
53 r1suc 8240 . . . . . . . . . . . . . 14  |-  ( 1o  e.  On  ->  ( R1 `  suc  1o )  =  ~P ( R1
`  1o ) )
5452, 53ax-mp 5 . . . . . . . . . . . . 13  |-  ( R1
`  suc  1o )  =  ~P ( R1 `  1o )
5551, 54eqtr4i 2461 . . . . . . . . . . . 12  |-  { (/) ,  { (/) } }  =  ( R1 `  suc  1o )
5638, 55syl6eqr 2488 . . . . . . . . . . 11  |-  ( y  =  1o  ->  ( R1 `  suc  y )  =  { (/) ,  { (/)
} } )
5756eleq2d 2499 . . . . . . . . . 10  |-  ( y  =  1o  ->  (
x  e.  ( R1
`  suc  y )  <->  x  e.  { (/) ,  { (/)
} } ) )
5857elrab 3235 . . . . . . . . 9  |-  ( 1o  e.  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  <->  ( 1o  e.  On  /\  x  e. 
{ (/) ,  { (/) } } ) )
5936, 58sylib 199 . . . . . . . 8  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  ( 1o  e.  On  /\  x  e.  { (/) ,  { (/) } } ) )
6013elpr 4020 . . . . . . . . . 10  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
61 df-ne 2627 . . . . . . . . . . . 12  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
62 orel1 383 . . . . . . . . . . . 12  |-  ( -.  x  =  (/)  ->  (
( x  =  (/)  \/  x  =  { (/) } )  ->  x  =  { (/) } ) )
6361, 62sylbi 198 . . . . . . . . . . 11  |-  ( x  =/=  (/)  ->  ( (
x  =  (/)  \/  x  =  { (/) } )  ->  x  =  { (/) } ) )
64 eqeq2 2444 . . . . . . . . . . . . 13  |-  ( x  =  { (/) }  ->  ( 1o  =  x  <->  1o  =  { (/) } ) )
6521, 64mpbiri 236 . . . . . . . . . . . 12  |-  ( x  =  { (/) }  ->  1o  =  x )
6665eqcomd 2437 . . . . . . . . . . 11  |-  ( x  =  { (/) }  ->  x  =  1o )
6763, 66syl6com 36 . . . . . . . . . 10  |-  ( ( x  =  (/)  \/  x  =  { (/) } )  -> 
( x  =/=  (/)  ->  x  =  1o ) )
6860, 67sylbi 198 . . . . . . . . 9  |-  ( x  e.  { (/) ,  { (/)
} }  ->  (
x  =/=  (/)  ->  x  =  1o ) )
6968adantl 467 . . . . . . . 8  |-  ( ( 1o  e.  On  /\  x  e.  { (/) ,  { (/)
} } )  -> 
( x  =/=  (/)  ->  x  =  1o ) )
7059, 69syl 17 . . . . . . 7  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  (
x  =/=  (/)  ->  x  =  1o ) )
7118, 70sylbi 198 . . . . . 6  |-  ( (
rank `  x )  =  1o  ->  ( x  =/=  (/)  ->  x  =  1o ) )
7216, 71mpd 15 . . . . 5  |-  ( (
rank `  x )  =  1o  ->  x  =  1o )
7310, 72vtoclg 3145 . . . 4  |-  ( A  e.  _V  ->  (
( rank `  A )  =  1o  ->  A  =  1o ) )
746, 73mpcom 37 . . 3  |-  ( (
rank `  A )  =  1o  ->  A  =  1o )
75 fveq2 5881 . . . 4  |-  ( A  =  1o  ->  ( rank `  A )  =  ( rank `  1o ) )
76 r111 8245 . . . . . . 7  |-  R1 : On
-1-1-> _V
77 f1dm 5800 . . . . . . 7  |-  ( R1 : On -1-1-> _V  ->  dom 
R1  =  On )
7876, 77ax-mp 5 . . . . . 6  |-  dom  R1  =  On
7952, 78eleqtrri 2516 . . . . 5  |-  1o  e.  dom  R1
80 rankonid 8299 . . . . 5  |-  ( 1o  e.  dom  R1  <->  ( rank `  1o )  =  1o )
8179, 80mpbi 211 . . . 4  |-  ( rank `  1o )  =  1o
8275, 81syl6eq 2486 . . 3  |-  ( A  =  1o  ->  ( rank `  A )  =  1o )
8374, 82impbii 190 . 2  |-  ( (
rank `  A )  =  1o  <->  A  =  1o )
8421eqeq2i 2447 . 2  |-  ( A  =  1o  <->  A  =  { (/) } )
8583, 84bitri 252 1  |-  ( (
rank `  A )  =  1o  <->  A  =  { (/)
} )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   {crab 2786   _Vcvv 3087    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   {csn 4002   {cpr 4004   |^|cint 4258   dom cdm 4854   Oncon0 5442   suc csuc 5444   -1-1->wf1 5598   ` cfv 5601   1oc1o 7183   R1cr1 8232   rankcrnk 8233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-reg 8107  ax-inf2 8146
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-r1 8234  df-rank 8235
This theorem is referenced by: (None)
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