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Theorem r1tr 8095
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 8085 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 462 . . . . 5  |-  Lim  dom  R1
3 limord 4887 . . . . 5  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordsson 6512 . . . . 5  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
52, 3, 4mp2b 10 . . . 4  |-  dom  R1  C_  On
65sseli 3461 . . 3  |-  ( A  e.  dom  R1  ->  A  e.  On )
7 fveq2 5800 . . . . . 6  |-  ( x  =  (/)  ->  ( R1
`  x )  =  ( R1 `  (/) ) )
8 r10 8087 . . . . . 6  |-  ( R1
`  (/) )  =  (/)
97, 8syl6eq 2511 . . . . 5  |-  ( x  =  (/)  ->  ( R1
`  x )  =  (/) )
10 treq 4500 . . . . 5  |-  ( ( R1 `  x )  =  (/)  ->  ( Tr  ( R1 `  x
)  <->  Tr  (/) ) )
119, 10syl 16 . . . 4  |-  ( x  =  (/)  ->  ( Tr  ( R1 `  x
)  <->  Tr  (/) ) )
12 fveq2 5800 . . . . 5  |-  ( x  =  y  ->  ( R1 `  x )  =  ( R1 `  y
) )
13 treq 4500 . . . . 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 5800 . . . . 5  |-  ( x  =  suc  y  -> 
( R1 `  x
)  =  ( R1
`  suc  y )
)
16 treq 4500 . . . . 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 5800 . . . . 5  |-  ( x  =  A  ->  ( R1 `  x )  =  ( R1 `  A
) )
19 treq 4500 . . . . 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 4505 . . . 4  |-  Tr  (/)
22 limsuc 6571 . . . . . . . 8  |-  ( Lim 
dom  R1  ->  ( y  e.  dom  R1  <->  suc  y  e. 
dom  R1 ) )
232, 22ax-mp 5 . . . . . . 7  |-  ( y  e.  dom  R1  <->  suc  y  e. 
dom  R1 )
24 simpr 461 . . . . . . . . 9  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  Tr  ( R1 `  y ) )
25 pwtr 4654 . . . . . . . . 9  |-  ( Tr  ( R1 `  y
)  <->  Tr  ~P ( R1 `  y ) )
2624, 25sylib 196 . . . . . . . 8  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  Tr  ~P ( R1 `  y
) )
27 r1sucg 8088 . . . . . . . . 9  |-  ( y  e.  dom  R1  ->  ( R1 `  suc  y
)  =  ~P ( R1 `  y ) )
28 treq 4500 . . . . . . . . 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 222 . . . . . . 7  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  (
y  e.  dom  R1  ->  Tr  ( R1 `  suc  y ) ) )
3123, 30syl5bir 218 . . . . . 6  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  ( suc  y  e.  dom  R1 
->  Tr  ( R1 `  suc  y ) ) )
32 ndmfv 5824 . . . . . . . 8  |-  ( -. 
suc  y  e.  dom  R1 
->  ( R1 `  suc  y )  =  (/) )
33 treq 4500 . . . . . . . 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 233 . . . . . 6  |-  ( -. 
suc  y  e.  dom  R1 
->  Tr  ( R1 `  suc  y ) )
3631, 35pm2.61d1 159 . . . . 5  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  Tr  ( R1 `  suc  y
) )
3736ex 434 . . . 4  |-  ( y  e.  On  ->  ( Tr  ( R1 `  y
)  ->  Tr  ( R1 `  suc  y ) ) )
38 triun 4507 . . . . . . . 8  |-  ( A. y  e.  x  Tr  ( R1 `  y )  ->  Tr  U_ y  e.  x  ( R1 `  y ) )
39 r1limg 8090 . . . . . . . . . 10  |-  ( ( x  e.  dom  R1  /\ 
Lim  x )  -> 
( R1 `  x
)  =  U_ y  e.  x  ( R1 `  y ) )
4039ancoms 453 . . . . . . . . 9  |-  ( ( Lim  x  /\  x  e.  dom  R1 )  -> 
( R1 `  x
)  =  U_ y  e.  x  ( R1 `  y ) )
41 treq 4500 . . . . . . . . 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 221 . . . . . . 7  |-  ( ( Lim  x  /\  x  e.  dom  R1 )  -> 
( A. y  e.  x  Tr  ( R1
`  y )  ->  Tr  ( R1 `  x
) ) )
4443impancom 440 . . . . . 6  |-  ( ( Lim  x  /\  A. y  e.  x  Tr  ( R1 `  y ) )  ->  ( x  e.  dom  R1  ->  Tr  ( R1 `  x ) ) )
45 ndmfv 5824 . . . . . . . 8  |-  ( -.  x  e.  dom  R1  ->  ( R1 `  x
)  =  (/) )
4645, 10syl 16 . . . . . . 7  |-  ( -.  x  e.  dom  R1  ->  ( Tr  ( R1
`  x )  <->  Tr  (/) ) )
4721, 46mpbiri 233 . . . . . 6  |-  ( -.  x  e.  dom  R1  ->  Tr  ( R1 `  x ) )
4844, 47pm2.61d1 159 . . . . 5  |-  ( ( Lim  x  /\  A. y  e.  x  Tr  ( R1 `  y ) )  ->  Tr  ( R1 `  x ) )
4948ex 434 . . . 4  |-  ( Lim  x  ->  ( A. y  e.  x  Tr  ( R1 `  y )  ->  Tr  ( R1 `  x ) ) )
5011, 14, 17, 20, 21, 37, 49tfinds 6581 . . 3  |-  ( A  e.  On  ->  Tr  ( R1 `  A ) )
516, 50syl 16 . 2  |-  ( A  e.  dom  R1  ->  Tr  ( R1 `  A
) )
52 ndmfv 5824 . . . 4  |-  ( -.  A  e.  dom  R1  ->  ( R1 `  A
)  =  (/) )
53 treq 4500 . . . 4  |-  ( ( R1 `  A )  =  (/)  ->  ( Tr  ( R1 `  A
)  <->  Tr  (/) ) )
5452, 53syl 16 . . 3  |-  ( -.  A  e.  dom  R1  ->  ( Tr  ( R1
`  A )  <->  Tr  (/) ) )
5521, 54mpbiri 233 . 2  |-  ( -.  A  e.  dom  R1  ->  Tr  ( R1 `  A ) )
5651, 55pm2.61i 164 1  |-  Tr  ( R1 `  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799    C_ wss 3437   (/)c0 3746   ~Pcpw 3969   U_ciun 4280   Tr wtr 4494   Ord word 4827   Oncon0 4828   Lim wlim 4829   suc csuc 4830   dom cdm 4949   Fun wfun 5521   ` cfv 5527   R1cr1 8081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-om 6588  df-recs 6943  df-rdg 6977  df-r1 8083
This theorem is referenced by:  r1tr2  8096  r1ordg  8097  r1ord3g  8098  r1ord2  8100  r1sssuc  8102  r1pwss  8103  r1val1  8105  rankwflemb  8112  r1elwf  8115  r1elssi  8124  uniwf  8138  tcrank  8203  ackbij2lem3  8522  r1limwun  9015  tskr1om2  9047
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