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Theorem r1tr 8211
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 8201 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 462 . . . . 5  |-  Lim  dom  R1
3 limord 4946 . . . . 5  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordsson 6624 . . . . 5  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
52, 3, 4mp2b 10 . . . 4  |-  dom  R1  C_  On
65sseli 3495 . . 3  |-  ( A  e.  dom  R1  ->  A  e.  On )
7 fveq2 5872 . . . . . 6  |-  ( x  =  (/)  ->  ( R1
`  x )  =  ( R1 `  (/) ) )
8 r10 8203 . . . . . 6  |-  ( R1
`  (/) )  =  (/)
97, 8syl6eq 2514 . . . . 5  |-  ( x  =  (/)  ->  ( R1
`  x )  =  (/) )
10 treq 4556 . . . . 5  |-  ( ( R1 `  x )  =  (/)  ->  ( Tr  ( R1 `  x
)  <->  Tr  (/) ) )
119, 10syl 16 . . . 4  |-  ( x  =  (/)  ->  ( Tr  ( R1 `  x
)  <->  Tr  (/) ) )
12 fveq2 5872 . . . . 5  |-  ( x  =  y  ->  ( R1 `  x )  =  ( R1 `  y
) )
13 treq 4556 . . . . 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 5872 . . . . 5  |-  ( x  =  suc  y  -> 
( R1 `  x
)  =  ( R1
`  suc  y )
)
16 treq 4556 . . . . 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 5872 . . . . 5  |-  ( x  =  A  ->  ( R1 `  x )  =  ( R1 `  A
) )
19 treq 4556 . . . . 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 4561 . . . 4  |-  Tr  (/)
22 limsuc 6683 . . . . . . . 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 4709 . . . . . . . . 9  |-  ( Tr  ( R1 `  y
)  <->  Tr  ~P ( R1 `  y ) )
2624, 25sylib 196 . . . . . . . 8  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  Tr  ~P ( R1 `  y
) )
27 r1sucg 8204 . . . . . . . . 9  |-  ( y  e.  dom  R1  ->  ( R1 `  suc  y
)  =  ~P ( R1 `  y ) )
28 treq 4556 . . . . . . . . 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 5896 . . . . . . . 8  |-  ( -. 
suc  y  e.  dom  R1 
->  ( R1 `  suc  y )  =  (/) )
33 treq 4556 . . . . . . . 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 4563 . . . . . . . 8  |-  ( A. y  e.  x  Tr  ( R1 `  y )  ->  Tr  U_ y  e.  x  ( R1 `  y ) )
39 r1limg 8206 . . . . . . . . . 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 4556 . . . . . . . . 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 5896 . . . . . . . 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 6693 . . 3  |-  ( A  e.  On  ->  Tr  ( R1 `  A ) )
516, 50syl 16 . 2  |-  ( A  e.  dom  R1  ->  Tr  ( R1 `  A
) )
52 ndmfv 5896 . . . 4  |-  ( -.  A  e.  dom  R1  ->  ( R1 `  A
)  =  (/) )
53 treq 4556 . . . 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 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   U_ciun 4332   Tr wtr 4550   Ord word 4886   Oncon0 4887   Lim wlim 4888   suc csuc 4889   dom cdm 5008   Fun wfun 5588   ` cfv 5594   R1cr1 8197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-r1 8199
This theorem is referenced by:  r1tr2  8212  r1ordg  8213  r1ord3g  8214  r1ord2  8216  r1sssuc  8218  r1pwss  8219  r1val1  8221  rankwflemb  8228  r1elwf  8231  r1elssi  8240  uniwf  8254  tcrank  8319  ackbij2lem3  8638  r1limwun  9131  tskr1om2  9163
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