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Theorem rankeq1o 29391
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 7135 . . . . . . 7  |-  1o  =/=  (/)
2 neeq1 2741 . . . . . . 7  |-  ( (
rank `  A )  =  1o  ->  ( (
rank `  A )  =/=  (/)  <->  1o  =/=  (/) ) )
31, 2mpbiri 233 . . . . . 6  |-  ( (
rank `  A )  =  1o  ->  ( rank `  A )  =/=  (/) )
43neneqd 2662 . . . . 5  |-  ( (
rank `  A )  =  1o  ->  -.  ( rank `  A )  =  (/) )
5 fvprc 5851 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
64, 5nsyl2 127 . . . 4  |-  ( (
rank `  A )  =  1o  ->  A  e. 
_V )
7 fveq2 5857 . . . . . . 7  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
87eqeq1d 2462 . . . . . 6  |-  ( x  =  A  ->  (
( rank `  x )  =  1o  <->  ( rank `  A
)  =  1o ) )
9 eqeq1 2464 . . . . . 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 2741 . . . . . . . 8  |-  ( (
rank `  x )  =  1o  ->  ( (
rank `  x )  =/=  (/)  <->  1o  =/=  (/) ) )
121, 11mpbiri 233 . . . . . . 7  |-  ( (
rank `  x )  =  1o  ->  ( rank `  x )  =/=  (/) )
13 vex 3109 . . . . . . . . 9  |-  x  e. 
_V
1413rankeq0 8268 . . . . . . . 8  |-  ( x  =  (/)  <->  ( rank `  x
)  =  (/) )
1514necon3bii 2728 . . . . . . 7  |-  ( x  =/=  (/)  <->  ( rank `  x
)  =/=  (/) )
1612, 15sylibr 212 . . . . . 6  |-  ( (
rank `  x )  =  1o  ->  x  =/=  (/) )
1713rankval 8223 . . . . . . . 8  |-  ( rank `  x )  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }
1817eqeq1i 2467 . . . . . . 7  |-  ( (
rank `  x )  =  1o  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  1o )
19 ssrab2 3578 . . . . . . . . . . 11  |-  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  C_  On
20 elirr 8013 . . . . . . . . . . . . . 14  |-  -.  1o  e.  1o
21 df1o2 7132 . . . . . . . . . . . . . . . 16  |-  1o  =  { (/) }
22 p0ex 4627 . . . . . . . . . . . . . . . 16  |-  { (/) }  e.  _V
2321, 22eqeltri 2544 . . . . . . . . . . . . . . 15  |-  1o  e.  _V
24 id 22 . . . . . . . . . . . . . . 15  |-  ( _V  =  1o  ->  _V  =  1o )
2523, 24syl5eleq 2554 . . . . . . . . . . . . . 14  |-  ( _V  =  1o  ->  1o  e.  1o )
2620, 25mto 176 . . . . . . . . . . . . 13  |-  -.  _V  =  1o
27 inteq 4278 . . . . . . . . . . . . . . 15  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  |^| (/) )
28 int0 4289 . . . . . . . . . . . . . . 15  |-  |^| (/)  =  _V
2927, 28syl6eq 2517 . . . . . . . . . . . . . 14  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  _V )
3029eqeq1d 2462 . . . . . . . . . . . . 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 2695 . . . . . . . . . . 11  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =/=  (/) )
33 onint 6601 . . . . . . . . . . 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 2532 . . . . . . . . . 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 4936 . . . . . . . . . . . . 13  |-  ( y  =  1o  ->  suc  y  =  suc  1o )
3837fveq2d 5861 . . . . . . . . . . . 12  |-  ( y  =  1o  ->  ( R1 `  suc  y )  =  ( R1 `  suc  1o ) )
39 df-1o 7120 . . . . . . . . . . . . . . . . 17  |-  1o  =  suc  (/)
4039fveq2i 5860 . . . . . . . . . . . . . . . 16  |-  ( R1
`  1o )  =  ( R1 `  suc  (/) )
41 0elon 4924 . . . . . . . . . . . . . . . . 17  |-  (/)  e.  On
42 r1suc 8177 . . . . . . . . . . . . . . . . 17  |-  ( (/)  e.  On  ->  ( R1 ` 
suc  (/) )  =  ~P ( R1 `  (/) ) )
4341, 42ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( R1
`  suc  (/) )  =  ~P ( R1 `  (/) )
44 r10 8175 . . . . . . . . . . . . . . . . 17  |-  ( R1
`  (/) )  =  (/)
4544pweqi 4007 . . . . . . . . . . . . . . . 16  |-  ~P ( R1 `  (/) )  =  ~P (/)
4640, 43, 453eqtri 2493 . . . . . . . . . . . . . . 15  |-  ( R1
`  1o )  =  ~P (/)
4746pweqi 4007 . . . . . . . . . . . . . 14  |-  ~P ( R1 `  1o )  =  ~P ~P (/)
48 pw0 4167 . . . . . . . . . . . . . . 15  |-  ~P (/)  =  { (/)
}
4948pweqi 4007 . . . . . . . . . . . . . 14  |-  ~P ~P (/)  =  ~P { (/) }
50 pwpw0 4168 . . . . . . . . . . . . . 14  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
5147, 49, 503eqtrri 2494 . . . . . . . . . . . . 13  |-  { (/) ,  { (/) } }  =  ~P ( R1 `  1o )
52 1on 7127 . . . . . . . . . . . . . 14  |-  1o  e.  On
53 r1suc 8177 . . . . . . . . . . . . . 14  |-  ( 1o  e.  On  ->  ( R1 `  suc  1o )  =  ~P ( R1
`  1o ) )
5452, 53ax-mp 5 . . . . . . . . . . . . 13  |-  ( R1
`  suc  1o )  =  ~P ( R1 `  1o )
5551, 54eqtr4i 2492 . . . . . . . . . . . 12  |-  { (/) ,  { (/) } }  =  ( R1 `  suc  1o )
5638, 55syl6eqr 2519 . . . . . . . . . . 11  |-  ( y  =  1o  ->  ( R1 `  suc  y )  =  { (/) ,  { (/)
} } )
5756eleq2d 2530 . . . . . . . . . 10  |-  ( y  =  1o  ->  (
x  e.  ( R1
`  suc  y )  <->  x  e.  { (/) ,  { (/)
} } ) )
5857elrab 3254 . . . . . . . . 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 4038 . . . . . . . . . 10  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
61 df-ne 2657 . . . . . . . . . . . 12  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
62 orel1 382 . . . . . . . . . . . 12  |-  ( -.  x  =  (/)  ->  (
( x  =  (/)  \/  x  =  { (/) } )  ->  x  =  { (/) } ) )
6361, 62sylbi 195 . . . . . . . . . . 11  |-  ( x  =/=  (/)  ->  ( (
x  =  (/)  \/  x  =  { (/) } )  ->  x  =  { (/) } ) )
64 eqeq2 2475 . . . . . . . . . . . . 13  |-  ( x  =  { (/) }  ->  ( 1o  =  x  <->  1o  =  { (/) } ) )
6521, 64mpbiri 233 . . . . . . . . . . . 12  |-  ( x  =  { (/) }  ->  1o  =  x )
6665eqcomd 2468 . . . . . . . . . . 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 3164 . . . 4  |-  ( A  e.  _V  ->  (
( rank `  A )  =  1o  ->  A  =  1o ) )
746, 73mpcom 36 . . 3  |-  ( (
rank `  A )  =  1o  ->  A  =  1o )
75 fveq2 5857 . . . 4  |-  ( A  =  1o  ->  ( rank `  A )  =  ( rank `  1o ) )
76 r111 8182 . . . . . . 7  |-  R1 : On
-1-1-> _V
77 f1dm 5776 . . . . . . 7  |-  ( R1 : On -1-1-> _V  ->  dom 
R1  =  On )
7876, 77ax-mp 5 . . . . . 6  |-  dom  R1  =  On
7952, 78eleqtrri 2547 . . . . 5  |-  1o  e.  dom  R1
80 rankonid 8236 . . . . 5  |-  ( 1o  e.  dom  R1  <->  ( rank `  1o )  =  1o )
8179, 80mpbi 208 . . . 4  |-  ( rank `  1o )  =  1o
8275, 81syl6eq 2517 . . 3  |-  ( A  =  1o  ->  ( rank `  A )  =  1o )
8374, 82impbii 188 . 2  |-  ( (
rank `  A )  =  1o  <->  A  =  1o )
8421eqeq2i 2478 . 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 1374    e. wcel 1762    =/= wne 2655   {crab 2811   _Vcvv 3106    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   {csn 4020   {cpr 4022   |^|cint 4275   Oncon0 4871   suc csuc 4873   dom cdm 4992   -1-1->wf1 5576   ` cfv 5579   1oc1o 7113   R1cr1 8169   rankcrnk 8170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-reg 8007  ax-inf2 8047
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-recs 7032  df-rdg 7066  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-r1 8171  df-rank 8172
This theorem is referenced by: (None)
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