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Theorem rankeq1o 30949
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 7214 . . . . . . 7  |-  1o  =/=  (/)
2 neeq1 2706 . . . . . . 7  |-  ( (
rank `  A )  =  1o  ->  ( (
rank `  A )  =/=  (/)  <->  1o  =/=  (/) ) )
31, 2mpbiri 237 . . . . . 6  |-  ( (
rank `  A )  =  1o  ->  ( rank `  A )  =/=  (/) )
43neneqd 2626 . . . . 5  |-  ( (
rank `  A )  =  1o  ->  -.  ( rank `  A )  =  (/) )
5 fvprc 5881 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
64, 5nsyl2 131 . . . 4  |-  ( (
rank `  A )  =  1o  ->  A  e. 
_V )
7 fveq2 5887 . . . . . . 7  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
87eqeq1d 2425 . . . . . 6  |-  ( x  =  A  ->  (
( rank `  x )  =  1o  <->  ( rank `  A
)  =  1o ) )
9 eqeq1 2427 . . . . . 6  |-  ( x  =  A  ->  (
x  =  1o  <->  A  =  1o ) )
108, 9imbi12d 322 . . . . 5  |-  ( x  =  A  ->  (
( ( rank `  x
)  =  1o  ->  x  =  1o )  <->  ( ( rank `  A )  =  1o  ->  A  =  1o ) ) )
11 neeq1 2706 . . . . . . . 8  |-  ( (
rank `  x )  =  1o  ->  ( (
rank `  x )  =/=  (/)  <->  1o  =/=  (/) ) )
121, 11mpbiri 237 . . . . . . 7  |-  ( (
rank `  x )  =  1o  ->  ( rank `  x )  =/=  (/) )
13 vex 3088 . . . . . . . . 9  |-  x  e. 
_V
1413rankeq0 8346 . . . . . . . 8  |-  ( x  =  (/)  <->  ( rank `  x
)  =  (/) )
1514necon3bii 2693 . . . . . . 7  |-  ( x  =/=  (/)  <->  ( rank `  x
)  =/=  (/) )
1612, 15sylibr 216 . . . . . 6  |-  ( (
rank `  x )  =  1o  ->  x  =/=  (/) )
1713rankval 8301 . . . . . . . 8  |-  ( rank `  x )  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }
1817eqeq1i 2430 . . . . . . 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 8128 . . . . . . . . . . . . . 14  |-  -.  1o  e.  1o
21 df1o2 7211 . . . . . . . . . . . . . . . 16  |-  1o  =  { (/) }
22 p0ex 4617 . . . . . . . . . . . . . . . 16  |-  { (/) }  e.  _V
2321, 22eqeltri 2508 . . . . . . . . . . . . . . 15  |-  1o  e.  _V
24 id 23 . . . . . . . . . . . . . . 15  |-  ( _V  =  1o  ->  _V  =  1o )
2523, 24syl5eleq 2518 . . . . . . . . . . . . . 14  |-  ( _V  =  1o  ->  1o  e.  1o )
2620, 25mto 180 . . . . . . . . . . . . 13  |-  -.  _V  =  1o
27 inteq 4264 . . . . . . . . . . . . . . 15  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  |^| (/) )
28 int0 4275 . . . . . . . . . . . . . . 15  |-  |^| (/)  =  _V
2927, 28syl6eq 2480 . . . . . . . . . . . . . 14  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  _V )
3029eqeq1d 2425 . . . . . . . . . . . . 13  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  <->  _V  =  1o ) )
3126, 30mtbiri 305 . . . . . . . . . . . 12  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  -.  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o )
3231necon2ai 2660 . . . . . . . . . . 11  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =/=  (/) )
33 onint 6642 . . . . . . . . . . 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 668 . . . . . . . . . 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 2496 . . . . . . . . . 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 214 . . . . . . . . 9  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  1o  e.  { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
37 suceq 5513 . . . . . . . . . . . . 13  |-  ( y  =  1o  ->  suc  y  =  suc  1o )
3837fveq2d 5891 . . . . . . . . . . . 12  |-  ( y  =  1o  ->  ( R1 `  suc  y )  =  ( R1 `  suc  1o ) )
39 df-1o 7199 . . . . . . . . . . . . . . . . 17  |-  1o  =  suc  (/)
4039fveq2i 5890 . . . . . . . . . . . . . . . 16  |-  ( R1
`  1o )  =  ( R1 `  suc  (/) )
41 0elon 5501 . . . . . . . . . . . . . . . . 17  |-  (/)  e.  On
42 r1suc 8255 . . . . . . . . . . . . . . . . 17  |-  ( (/)  e.  On  ->  ( R1 ` 
suc  (/) )  =  ~P ( R1 `  (/) ) )
4341, 42ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( R1
`  suc  (/) )  =  ~P ( R1 `  (/) )
44 r10 8253 . . . . . . . . . . . . . . . . 17  |-  ( R1
`  (/) )  =  (/)
4544pweqi 3991 . . . . . . . . . . . . . . . 16  |-  ~P ( R1 `  (/) )  =  ~P (/)
4640, 43, 453eqtri 2456 . . . . . . . . . . . . . . 15  |-  ( R1
`  1o )  =  ~P (/)
4746pweqi 3991 . . . . . . . . . . . . . 14  |-  ~P ( R1 `  1o )  =  ~P ~P (/)
48 pw0 4153 . . . . . . . . . . . . . . 15  |-  ~P (/)  =  { (/)
}
4948pweqi 3991 . . . . . . . . . . . . . 14  |-  ~P ~P (/)  =  ~P { (/) }
50 pwpw0 4154 . . . . . . . . . . . . . 14  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
5147, 49, 503eqtrri 2457 . . . . . . . . . . . . 13  |-  { (/) ,  { (/) } }  =  ~P ( R1 `  1o )
52 1on 7206 . . . . . . . . . . . . . 14  |-  1o  e.  On
53 r1suc 8255 . . . . . . . . . . . . . 14  |-  ( 1o  e.  On  ->  ( R1 `  suc  1o )  =  ~P ( R1
`  1o ) )
5452, 53ax-mp 5 . . . . . . . . . . . . 13  |-  ( R1
`  suc  1o )  =  ~P ( R1 `  1o )
5551, 54eqtr4i 2455 . . . . . . . . . . . 12  |-  { (/) ,  { (/) } }  =  ( R1 `  suc  1o )
5638, 55syl6eqr 2482 . . . . . . . . . . 11  |-  ( y  =  1o  ->  ( R1 `  suc  y )  =  { (/) ,  { (/)
} } )
5756eleq2d 2493 . . . . . . . . . 10  |-  ( y  =  1o  ->  (
x  e.  ( R1
`  suc  y )  <->  x  e.  { (/) ,  { (/)
} } ) )
5857elrab 3234 . . . . . . . . 9  |-  ( 1o  e.  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  <->  ( 1o  e.  On  /\  x  e. 
{ (/) ,  { (/) } } ) )
5936, 58sylib 200 . . . . . . . 8  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  ( 1o  e.  On  /\  x  e.  { (/) ,  { (/) } } ) )
6013elpr 4022 . . . . . . . . . 10  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
61 df-ne 2621 . . . . . . . . . . . 12  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
62 orel1 384 . . . . . . . . . . . 12  |-  ( -.  x  =  (/)  ->  (
( x  =  (/)  \/  x  =  { (/) } )  ->  x  =  { (/) } ) )
6361, 62sylbi 199 . . . . . . . . . . 11  |-  ( x  =/=  (/)  ->  ( (
x  =  (/)  \/  x  =  { (/) } )  ->  x  =  { (/) } ) )
64 eqeq2 2438 . . . . . . . . . . . . 13  |-  ( x  =  { (/) }  ->  ( 1o  =  x  <->  1o  =  { (/) } ) )
6521, 64mpbiri 237 . . . . . . . . . . . 12  |-  ( x  =  { (/) }  ->  1o  =  x )
6665eqcomd 2431 . . . . . . . . . . 11  |-  ( x  =  { (/) }  ->  x  =  1o )
6763, 66syl6com 37 . . . . . . . . . 10  |-  ( ( x  =  (/)  \/  x  =  { (/) } )  -> 
( x  =/=  (/)  ->  x  =  1o ) )
6860, 67sylbi 199 . . . . . . . . 9  |-  ( x  e.  { (/) ,  { (/)
} }  ->  (
x  =/=  (/)  ->  x  =  1o ) )
6968adantl 468 . . . . . . . 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 199 . . . . . 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 38 . . 3  |-  ( (
rank `  A )  =  1o  ->  A  =  1o )
75 fveq2 5887 . . . 4  |-  ( A  =  1o  ->  ( rank `  A )  =  ( rank `  1o ) )
76 r111 8260 . . . . . . 7  |-  R1 : On
-1-1-> _V
77 f1dm 5806 . . . . . . 7  |-  ( R1 : On -1-1-> _V  ->  dom 
R1  =  On )
7876, 77ax-mp 5 . . . . . 6  |-  dom  R1  =  On
7952, 78eleqtrri 2511 . . . . 5  |-  1o  e.  dom  R1
80 rankonid 8314 . . . . 5  |-  ( 1o  e.  dom  R1  <->  ( rank `  1o )  =  1o )
8179, 80mpbi 212 . . . 4  |-  ( rank `  1o )  =  1o
8275, 81syl6eq 2480 . . 3  |-  ( A  =  1o  ->  ( rank `  A )  =  1o )
8374, 82impbii 191 . 2  |-  ( (
rank `  A )  =  1o  <->  A  =  1o )
8421eqeq2i 2441 . 2  |-  ( A  =  1o  <->  A  =  { (/) } )
8583, 84bitri 253 1  |-  ( (
rank `  A )  =  1o  <->  A  =  { (/)
} )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1438    e. wcel 1873    =/= wne 2619   {crab 2780   _Vcvv 3085    C_ wss 3442   (/)c0 3767   ~Pcpw 3987   {csn 4004   {cpr 4006   |^|cint 4261   dom cdm 4859   Oncon0 5448   suc csuc 5450   -1-1->wf1 5604   ` cfv 5607   1oc1o 7192   R1cr1 8247   rankcrnk 8248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-rep 4542  ax-sep 4552  ax-nul 4561  ax-pow 4608  ax-pr 4666  ax-un 6603  ax-reg 8122  ax-inf2 8161
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3087  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 3918  df-pw 3989  df-sn 4005  df-pr 4007  df-tp 4009  df-op 4011  df-uni 4226  df-int 4262  df-iun 4307  df-br 4430  df-opab 4489  df-mpt 4490  df-tr 4525  df-eprel 4770  df-id 4774  df-po 4780  df-so 4781  df-fr 4818  df-we 4820  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-pred 5405  df-ord 5451  df-on 5452  df-lim 5453  df-suc 5454  df-iota 5571  df-fun 5609  df-fn 5610  df-f 5611  df-f1 5612  df-fo 5613  df-f1o 5614  df-fv 5615  df-om 6713  df-wrecs 7045  df-recs 7107  df-rdg 7145  df-1o 7199  df-er 7380  df-en 7587  df-dom 7588  df-sdom 7589  df-r1 8249  df-rank 8250
This theorem is referenced by: (None)
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