MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r1tr Unicode version

Theorem r1tr 7658
Description: The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
r1tr  |-  Tr  ( R1 `  A )

Proof of Theorem r1tr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1funlim 7648 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 449 . . . . 5  |-  Lim  dom  R1
3 limord 4600 . . . . 5  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordsson 4729 . . . . 5  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
52, 3, 4mp2b 10 . . . 4  |-  dom  R1  C_  On
65sseli 3304 . . 3  |-  ( A  e.  dom  R1  ->  A  e.  On )
7 fveq2 5687 . . . . . 6  |-  ( x  =  (/)  ->  ( R1
`  x )  =  ( R1 `  (/) ) )
8 r10 7650 . . . . . 6  |-  ( R1
`  (/) )  =  (/)
97, 8syl6eq 2452 . . . . 5  |-  ( x  =  (/)  ->  ( R1
`  x )  =  (/) )
10 treq 4268 . . . . 5  |-  ( ( R1 `  x )  =  (/)  ->  ( Tr  ( R1 `  x
)  <->  Tr  (/) ) )
119, 10syl 16 . . . 4  |-  ( x  =  (/)  ->  ( Tr  ( R1 `  x
)  <->  Tr  (/) ) )
12 fveq2 5687 . . . . 5  |-  ( x  =  y  ->  ( R1 `  x )  =  ( R1 `  y
) )
13 treq 4268 . . . . 5  |-  ( ( R1 `  x )  =  ( R1 `  y )  ->  ( Tr  ( R1 `  x
)  <->  Tr  ( R1 `  y ) ) )
1412, 13syl 16 . . . 4  |-  ( x  =  y  ->  ( Tr  ( R1 `  x
)  <->  Tr  ( R1 `  y ) ) )
15 fveq2 5687 . . . . 5  |-  ( x  =  suc  y  -> 
( R1 `  x
)  =  ( R1
`  suc  y )
)
16 treq 4268 . . . . 5  |-  ( ( R1 `  x )  =  ( R1 `  suc  y )  ->  ( Tr  ( R1 `  x
)  <->  Tr  ( R1 ` 
suc  y ) ) )
1715, 16syl 16 . . . 4  |-  ( x  =  suc  y  -> 
( Tr  ( R1
`  x )  <->  Tr  ( R1 `  suc  y ) ) )
18 fveq2 5687 . . . . 5  |-  ( x  =  A  ->  ( R1 `  x )  =  ( R1 `  A
) )
19 treq 4268 . . . . 5  |-  ( ( R1 `  x )  =  ( R1 `  A )  ->  ( Tr  ( R1 `  x
)  <->  Tr  ( R1 `  A ) ) )
2018, 19syl 16 . . . 4  |-  ( x  =  A  ->  ( Tr  ( R1 `  x
)  <->  Tr  ( R1 `  A ) ) )
21 tr0 4273 . . . 4  |-  Tr  (/)
22 limsuc 4788 . . . . . . . 8  |-  ( Lim 
dom  R1  ->  ( y  e.  dom  R1  <->  suc  y  e. 
dom  R1 ) )
232, 22ax-mp 8 . . . . . . 7  |-  ( y  e.  dom  R1  <->  suc  y  e. 
dom  R1 )
24 simpr 448 . . . . . . . . 9  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  Tr  ( R1 `  y ) )
25 pwtr 4376 . . . . . . . . 9  |-  ( Tr  ( R1 `  y
)  <->  Tr  ~P ( R1 `  y ) )
2624, 25sylib 189 . . . . . . . 8  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  Tr  ~P ( R1 `  y
) )
27 r1sucg 7651 . . . . . . . . 9  |-  ( y  e.  dom  R1  ->  ( R1 `  suc  y
)  =  ~P ( R1 `  y ) )
28 treq 4268 . . . . . . . . 9  |-  ( ( R1 `  suc  y
)  =  ~P ( R1 `  y )  -> 
( Tr  ( R1
`  suc  y )  <->  Tr 
~P ( R1 `  y ) ) )
2927, 28syl 16 . . . . . . . 8  |-  ( y  e.  dom  R1  ->  ( Tr  ( R1 `  suc  y )  <->  Tr  ~P ( R1 `  y ) ) )
3026, 29syl5ibrcom 214 . . . . . . 7  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  (
y  e.  dom  R1  ->  Tr  ( R1 `  suc  y ) ) )
3123, 30syl5bir 210 . . . . . 6  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  ( suc  y  e.  dom  R1 
->  Tr  ( R1 `  suc  y ) ) )
32 ndmfv 5714 . . . . . . . 8  |-  ( -. 
suc  y  e.  dom  R1 
->  ( R1 `  suc  y )  =  (/) )
33 treq 4268 . . . . . . . 8  |-  ( ( R1 `  suc  y
)  =  (/)  ->  ( Tr  ( R1 `  suc  y )  <->  Tr  (/) ) )
3432, 33syl 16 . . . . . . 7  |-  ( -. 
suc  y  e.  dom  R1 
->  ( Tr  ( R1
`  suc  y )  <->  Tr  (/) ) )
3521, 34mpbiri 225 . . . . . 6  |-  ( -. 
suc  y  e.  dom  R1 
->  Tr  ( R1 `  suc  y ) )
3631, 35pm2.61d1 153 . . . . 5  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  Tr  ( R1 `  suc  y
) )
3736ex 424 . . . 4  |-  ( y  e.  On  ->  ( Tr  ( R1 `  y
)  ->  Tr  ( R1 `  suc  y ) ) )
38 triun 4275 . . . . . . . 8  |-  ( A. y  e.  x  Tr  ( R1 `  y )  ->  Tr  U_ y  e.  x  ( R1 `  y ) )
39 r1limg 7653 . . . . . . . . . 10  |-  ( ( x  e.  dom  R1  /\ 
Lim  x )  -> 
( R1 `  x
)  =  U_ y  e.  x  ( R1 `  y ) )
4039ancoms 440 . . . . . . . . 9  |-  ( ( Lim  x  /\  x  e.  dom  R1 )  -> 
( R1 `  x
)  =  U_ y  e.  x  ( R1 `  y ) )
41 treq 4268 . . . . . . . . 9  |-  ( ( R1 `  x )  =  U_ y  e.  x  ( R1 `  y )  ->  ( Tr  ( R1 `  x
)  <->  Tr  U_ y  e.  x  ( R1 `  y ) ) )
4240, 41syl 16 . . . . . . . 8  |-  ( ( Lim  x  /\  x  e.  dom  R1 )  -> 
( Tr  ( R1
`  x )  <->  Tr  U_ y  e.  x  ( R1 `  y ) ) )
4338, 42syl5ibr 213 . . . . . . 7  |-  ( ( Lim  x  /\  x  e.  dom  R1 )  -> 
( A. y  e.  x  Tr  ( R1
`  y )  ->  Tr  ( R1 `  x
) ) )
4443impancom 428 . . . . . 6  |-  ( ( Lim  x  /\  A. y  e.  x  Tr  ( R1 `  y ) )  ->  ( x  e.  dom  R1  ->  Tr  ( R1 `  x ) ) )
45 ndmfv 5714 . . . . . . . 8  |-  ( -.  x  e.  dom  R1  ->  ( R1 `  x
)  =  (/) )
4645, 10syl 16 . . . . . . 7  |-  ( -.  x  e.  dom  R1  ->  ( Tr  ( R1
`  x )  <->  Tr  (/) ) )
4721, 46mpbiri 225 . . . . . 6  |-  ( -.  x  e.  dom  R1  ->  Tr  ( R1 `  x ) )
4844, 47pm2.61d1 153 . . . . 5  |-  ( ( Lim  x  /\  A. y  e.  x  Tr  ( R1 `  y ) )  ->  Tr  ( R1 `  x ) )
4948ex 424 . . . 4  |-  ( Lim  x  ->  ( A. y  e.  x  Tr  ( R1 `  y )  ->  Tr  ( R1 `  x ) ) )
5011, 14, 17, 20, 21, 37, 49tfinds 4798 . . 3  |-  ( A  e.  On  ->  Tr  ( R1 `  A ) )
516, 50syl 16 . 2  |-  ( A  e.  dom  R1  ->  Tr  ( R1 `  A
) )
52 ndmfv 5714 . . . 4  |-  ( -.  A  e.  dom  R1  ->  ( R1 `  A
)  =  (/) )
53 treq 4268 . . . 4  |-  ( ( R1 `  A )  =  (/)  ->  ( Tr  ( R1 `  A
)  <->  Tr  (/) ) )
5452, 53syl 16 . . 3  |-  ( -.  A  e.  dom  R1  ->  ( Tr  ( R1
`  A )  <->  Tr  (/) ) )
5521, 54mpbiri 225 . 2  |-  ( -.  A  e.  dom  R1  ->  Tr  ( R1 `  A ) )
5651, 55pm2.61i 158 1  |-  Tr  ( R1 `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   U_ciun 4053   Tr wtr 4262   Ord word 4540   Oncon0 4541   Lim wlim 4542   suc csuc 4543   dom cdm 4837   Fun wfun 5407   ` cfv 5413   R1cr1 7644
This theorem is referenced by:  r1tr2  7659  r1ordg  7660  r1ord3g  7661  r1ord2  7663  r1sssuc  7665  r1pwss  7666  r1val1  7668  rankwflemb  7675  r1elwf  7678  r1elssi  7687  uniwf  7701  tcrank  7764  ackbij2lem3  8077  r1limwun  8567  tskr1om2  8599
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-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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-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-r1 7646
  Copyright terms: Public domain W3C validator