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Theorem rankeq1o 26016
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 6698 . . . . . . 7  |-  1o  =/=  (/)
2 neeq1 2575 . . . . . . 7  |-  ( (
rank `  A )  =  1o  ->  ( (
rank `  A )  =/=  (/)  <->  1o  =/=  (/) ) )
31, 2mpbiri 225 . . . . . 6  |-  ( (
rank `  A )  =  1o  ->  ( rank `  A )  =/=  (/) )
43neneqd 2583 . . . . 5  |-  ( (
rank `  A )  =  1o  ->  -.  ( rank `  A )  =  (/) )
5 fvprc 5681 . . . . 5  |-  ( -.  A  e.  _V  ->  (
rank `  A )  =  (/) )
64, 5nsyl2 121 . . . 4  |-  ( (
rank `  A )  =  1o  ->  A  e. 
_V )
7 fveq2 5687 . . . . . . 7  |-  ( x  =  A  ->  ( rank `  x )  =  ( rank `  A
) )
87eqeq1d 2412 . . . . . 6  |-  ( x  =  A  ->  (
( rank `  x )  =  1o  <->  ( rank `  A
)  =  1o ) )
9 eqeq1 2410 . . . . . 6  |-  ( x  =  A  ->  (
x  =  1o  <->  A  =  1o ) )
108, 9imbi12d 312 . . . . 5  |-  ( x  =  A  ->  (
( ( rank `  x
)  =  1o  ->  x  =  1o )  <->  ( ( rank `  A )  =  1o  ->  A  =  1o ) ) )
11 neeq1 2575 . . . . . . . 8  |-  ( (
rank `  x )  =  1o  ->  ( (
rank `  x )  =/=  (/)  <->  1o  =/=  (/) ) )
121, 11mpbiri 225 . . . . . . 7  |-  ( (
rank `  x )  =  1o  ->  ( rank `  x )  =/=  (/) )
13 vex 2919 . . . . . . . . 9  |-  x  e. 
_V
1413rankeq0 7743 . . . . . . . 8  |-  ( x  =  (/)  <->  ( rank `  x
)  =  (/) )
1514necon3bii 2599 . . . . . . 7  |-  ( x  =/=  (/)  <->  ( rank `  x
)  =/=  (/) )
1612, 15sylibr 204 . . . . . 6  |-  ( (
rank `  x )  =  1o  ->  x  =/=  (/) )
1713rankval 7698 . . . . . . . 8  |-  ( rank `  x )  =  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }
1817eqeq1i 2411 . . . . . . 7  |-  ( (
rank `  x )  =  1o  <->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  1o )
19 ssrab2 3388 . . . . . . . . . . 11  |-  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  C_  On
20 elirr 7522 . . . . . . . . . . . . . 14  |-  -.  1o  e.  1o
21 df1o2 6695 . . . . . . . . . . . . . . . 16  |-  1o  =  { (/) }
22 p0ex 4346 . . . . . . . . . . . . . . . 16  |-  { (/) }  e.  _V
2321, 22eqeltri 2474 . . . . . . . . . . . . . . 15  |-  1o  e.  _V
24 id 20 . . . . . . . . . . . . . . 15  |-  ( _V  =  1o  ->  _V  =  1o )
2523, 24syl5eleq 2490 . . . . . . . . . . . . . 14  |-  ( _V  =  1o  ->  1o  e.  1o )
2620, 25mto 169 . . . . . . . . . . . . 13  |-  -.  _V  =  1o
27 inteq 4013 . . . . . . . . . . . . . . 15  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  |^| (/) )
28 int0 4024 . . . . . . . . . . . . . . 15  |-  |^| (/)  =  _V
2927, 28syl6eq 2452 . . . . . . . . . . . . . 14  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =  _V )
3029eqeq1d 2412 . . . . . . . . . . . . 13  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  <->  _V  =  1o ) )
3126, 30mtbiri 295 . . . . . . . . . . . 12  |-  ( { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  (/)  ->  -.  |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o )
3231necon2ai 2612 . . . . . . . . . . 11  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  =/=  (/) )
33 onint 4734 . . . . . . . . . . 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 645 . . . . . . . . . 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 2464 . . . . . . . . . 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 202 . . . . . . . . 9  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  1o  e.  { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
37 suceq 4606 . . . . . . . . . . . . 13  |-  ( y  =  1o  ->  suc  y  =  suc  1o )
3837fveq2d 5691 . . . . . . . . . . . 12  |-  ( y  =  1o  ->  ( R1 `  suc  y )  =  ( R1 `  suc  1o ) )
39 df-1o 6683 . . . . . . . . . . . . . . . . 17  |-  1o  =  suc  (/)
4039fveq2i 5690 . . . . . . . . . . . . . . . 16  |-  ( R1
`  1o )  =  ( R1 `  suc  (/) )
41 0elon 4594 . . . . . . . . . . . . . . . . 17  |-  (/)  e.  On
42 r1suc 7652 . . . . . . . . . . . . . . . . 17  |-  ( (/)  e.  On  ->  ( R1 ` 
suc  (/) )  =  ~P ( R1 `  (/) ) )
4341, 42ax-mp 8 . . . . . . . . . . . . . . . 16  |-  ( R1
`  suc  (/) )  =  ~P ( R1 `  (/) )
44 r10 7650 . . . . . . . . . . . . . . . . 17  |-  ( R1
`  (/) )  =  (/)
4544pweqi 3763 . . . . . . . . . . . . . . . 16  |-  ~P ( R1 `  (/) )  =  ~P (/)
4640, 43, 453eqtri 2428 . . . . . . . . . . . . . . 15  |-  ( R1
`  1o )  =  ~P (/)
4746pweqi 3763 . . . . . . . . . . . . . 14  |-  ~P ( R1 `  1o )  =  ~P ~P (/)
48 pw0 3905 . . . . . . . . . . . . . . 15  |-  ~P (/)  =  { (/)
}
4948pweqi 3763 . . . . . . . . . . . . . 14  |-  ~P ~P (/)  =  ~P { (/) }
50 pwpw0 3906 . . . . . . . . . . . . . 14  |-  ~P { (/)
}  =  { (/) ,  { (/) } }
5147, 49, 503eqtrri 2429 . . . . . . . . . . . . 13  |-  { (/) ,  { (/) } }  =  ~P ( R1 `  1o )
52 1on 6690 . . . . . . . . . . . . . 14  |-  1o  e.  On
53 r1suc 7652 . . . . . . . . . . . . . 14  |-  ( 1o  e.  On  ->  ( R1 `  suc  1o )  =  ~P ( R1
`  1o ) )
5452, 53ax-mp 8 . . . . . . . . . . . . 13  |-  ( R1
`  suc  1o )  =  ~P ( R1 `  1o )
5551, 54eqtr4i 2427 . . . . . . . . . . . 12  |-  { (/) ,  { (/) } }  =  ( R1 `  suc  1o )
5638, 55syl6eqr 2454 . . . . . . . . . . 11  |-  ( y  =  1o  ->  ( R1 `  suc  y )  =  { (/) ,  { (/)
} } )
5756eleq2d 2471 . . . . . . . . . 10  |-  ( y  =  1o  ->  (
x  e.  ( R1
`  suc  y )  <->  x  e.  { (/) ,  { (/)
} } ) )
5857elrab 3052 . . . . . . . . 9  |-  ( 1o  e.  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }  <->  ( 1o  e.  On  /\  x  e. 
{ (/) ,  { (/) } } ) )
5936, 58sylib 189 . . . . . . . 8  |-  ( |^| { y  e.  On  |  x  e.  ( R1 ` 
suc  y ) }  =  1o  ->  ( 1o  e.  On  /\  x  e.  { (/) ,  { (/) } } ) )
6013elpr 3792 . . . . . . . . . 10  |-  ( x  e.  { (/) ,  { (/)
} }  <->  ( x  =  (/)  \/  x  =  { (/) } ) )
61 df-ne 2569 . . . . . . . . . . . 12  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
62 orel1 372 . . . . . . . . . . . 12  |-  ( -.  x  =  (/)  ->  (
( x  =  (/)  \/  x  =  { (/) } )  ->  x  =  { (/) } ) )
6361, 62sylbi 188 . . . . . . . . . . 11  |-  ( x  =/=  (/)  ->  ( (
x  =  (/)  \/  x  =  { (/) } )  ->  x  =  { (/) } ) )
64 eqeq2 2413 . . . . . . . . . . . . 13  |-  ( x  =  { (/) }  ->  ( 1o  =  x  <->  1o  =  { (/) } ) )
6521, 64mpbiri 225 . . . . . . . . . . . 12  |-  ( x  =  { (/) }  ->  1o  =  x )
6665eqcomd 2409 . . . . . . . . . . 11  |-  ( x  =  { (/) }  ->  x  =  1o )
6763, 66syl6com 33 . . . . . . . . . 10  |-  ( ( x  =  (/)  \/  x  =  { (/) } )  -> 
( x  =/=  (/)  ->  x  =  1o ) )
6860, 67sylbi 188 . . . . . . . . 9  |-  ( x  e.  { (/) ,  { (/)
} }  ->  (
x  =/=  (/)  ->  x  =  1o ) )
6968adantl 453 . . . . . . . 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 188 . . . . . 6  |-  ( (
rank `  x )  =  1o  ->  ( x  =/=  (/)  ->  x  =  1o ) )
7216, 71mpd 15 . . . . 5  |-  ( (
rank `  x )  =  1o  ->  x  =  1o )
7310, 72vtoclg 2971 . . . 4  |-  ( A  e.  _V  ->  (
( rank `  A )  =  1o  ->  A  =  1o ) )
746, 73mpcom 34 . . 3  |-  ( (
rank `  A )  =  1o  ->  A  =  1o )
75 fveq2 5687 . . . 4  |-  ( A  =  1o  ->  ( rank `  A )  =  ( rank `  1o ) )
76 r111 7657 . . . . . . 7  |-  R1 : On
-1-1-> _V
77 f1dm 5602 . . . . . . 7  |-  ( R1 : On -1-1-> _V  ->  dom 
R1  =  On )
7876, 77ax-mp 8 . . . . . 6  |-  dom  R1  =  On
7952, 78eleqtrri 2477 . . . . 5  |-  1o  e.  dom  R1
80 rankonid 7711 . . . . 5  |-  ( 1o  e.  dom  R1  <->  ( rank `  1o )  =  1o )
8179, 80mpbi 200 . . . 4  |-  ( rank `  1o )  =  1o
8275, 81syl6eq 2452 . . 3  |-  ( A  =  1o  ->  ( rank `  A )  =  1o )
8374, 82impbii 181 . 2  |-  ( (
rank `  A )  =  1o  <->  A  =  1o )
8421eqeq2i 2414 . 2  |-  ( A  =  1o  <->  A  =  { (/) } )
8583, 84bitri 241 1  |-  ( (
rank `  A )  =  1o  <->  A  =  { (/)
} )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670   _Vcvv 2916    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   {cpr 3775   |^|cint 4010   Oncon0 4541   suc csuc 4543   dom cdm 4837   -1-1->wf1 5410   ` cfv 5413   1oc1o 6676   R1cr1 7644   rankcrnk 7645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-reg 7516  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-r1 7646  df-rank 7647
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