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Theorem r1tr 7416
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
StepHypRef Expression
1 r1funlim 7406 . . . . . 6  |-  ( Fun 
R1  /\  Lim  dom  R1 )
21simpri 450 . . . . 5  |-  Lim  dom  R1
3 limord 4423 . . . . 5  |-  ( Lim 
dom  R1  ->  Ord  dom  R1 )
4 ordsson 4553 . . . . 5  |-  ( Ord 
dom  R1  ->  dom  R1  C_  On )
52, 3, 4mp2b 11 . . . 4  |-  dom  R1  C_  On
65sseli 3151 . . 3  |-  ( A  e.  dom  R1  ->  A  e.  On )
7 fveq2 5458 . . . . . 6  |-  ( x  =  (/)  ->  ( R1
`  x )  =  ( R1 `  (/) ) )
8 r10 7408 . . . . . 6  |-  ( R1
`  (/) )  =  (/)
97, 8syl6eq 2306 . . . . 5  |-  ( x  =  (/)  ->  ( R1
`  x )  =  (/) )
10 treq 4093 . . . . 5  |-  ( ( R1 `  x )  =  (/)  ->  ( Tr  ( R1 `  x
)  <->  Tr  (/) ) )
119, 10syl 17 . . . 4  |-  ( x  =  (/)  ->  ( Tr  ( R1 `  x
)  <->  Tr  (/) ) )
12 fveq2 5458 . . . . 5  |-  ( x  =  y  ->  ( R1 `  x )  =  ( R1 `  y
) )
13 treq 4093 . . . . 5  |-  ( ( R1 `  x )  =  ( R1 `  y )  ->  ( Tr  ( R1 `  x
)  <->  Tr  ( R1 `  y ) ) )
1412, 13syl 17 . . . 4  |-  ( x  =  y  ->  ( Tr  ( R1 `  x
)  <->  Tr  ( R1 `  y ) ) )
15 fveq2 5458 . . . . 5  |-  ( x  =  suc  y  -> 
( R1 `  x
)  =  ( R1
`  suc  y )
)
16 treq 4093 . . . . 5  |-  ( ( R1 `  x )  =  ( R1 `  suc  y )  ->  ( Tr  ( R1 `  x
)  <->  Tr  ( R1 ` 
suc  y ) ) )
1715, 16syl 17 . . . 4  |-  ( x  =  suc  y  -> 
( Tr  ( R1
`  x )  <->  Tr  ( R1 `  suc  y ) ) )
18 fveq2 5458 . . . . 5  |-  ( x  =  A  ->  ( R1 `  x )  =  ( R1 `  A
) )
19 treq 4093 . . . . 5  |-  ( ( R1 `  x )  =  ( R1 `  A )  ->  ( Tr  ( R1 `  x
)  <->  Tr  ( R1 `  A ) ) )
2018, 19syl 17 . . . 4  |-  ( x  =  A  ->  ( Tr  ( R1 `  x
)  <->  Tr  ( R1 `  A ) ) )
21 tr0 4098 . . . 4  |-  Tr  (/)
22 limsuc 4612 . . . . . . . 8  |-  ( Lim 
dom  R1  ->  ( y  e.  dom  R1  <->  suc  y  e. 
dom  R1 ) )
232, 22ax-mp 10 . . . . . . 7  |-  ( y  e.  dom  R1  <->  suc  y  e. 
dom  R1 )
24 simpr 449 . . . . . . . . 9  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  Tr  ( R1 `  y ) )
25 pwtr 4198 . . . . . . . . 9  |-  ( Tr  ( R1 `  y
)  <->  Tr  ~P ( R1 `  y ) )
2624, 25sylib 190 . . . . . . . 8  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  Tr  ~P ( R1 `  y
) )
27 r1sucg 7409 . . . . . . . . 9  |-  ( y  e.  dom  R1  ->  ( R1 `  suc  y
)  =  ~P ( R1 `  y ) )
28 treq 4093 . . . . . . . . 9  |-  ( ( R1 `  suc  y
)  =  ~P ( R1 `  y )  -> 
( Tr  ( R1
`  suc  y )  <->  Tr 
~P ( R1 `  y ) ) )
2927, 28syl 17 . . . . . . . 8  |-  ( y  e.  dom  R1  ->  ( Tr  ( R1 `  suc  y )  <->  Tr  ~P ( R1 `  y ) ) )
3026, 29syl5ibrcom 215 . . . . . . 7  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  (
y  e.  dom  R1  ->  Tr  ( R1 `  suc  y ) ) )
3123, 30syl5bir 211 . . . . . 6  |-  ( ( y  e.  On  /\  Tr  ( R1 `  y
) )  ->  ( suc  y  e.  dom  R1 
->  Tr  ( R1 `  suc  y ) ) )
32 ndmfv 5486 . . . . . . . 8  |-  ( -. 
suc  y  e.  dom  R1 
->  ( R1 `  suc  y )  =  (/) )
33 treq 4093 . . . . . . . 8  |-  ( ( R1 `  suc  y
)  =  (/)  ->  ( Tr  ( R1 `  suc  y )  <->  Tr  (/) ) )
3432, 33syl 17 . . . . . . 7  |-  ( -. 
suc  y  e.  dom  R1 
->  ( Tr  ( R1
`  suc  y )  <->  Tr  (/) ) )
3521, 34mpbiri 226 . . . . . 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 425 . . . 4  |-  ( y  e.  On  ->  ( Tr  ( R1 `  y
)  ->  Tr  ( R1 `  suc  y ) ) )
38 triun 4100 . . . . . . . 8  |-  ( A. y  e.  x  Tr  ( R1 `  y )  ->  Tr  U_ y  e.  x  ( R1 `  y ) )
39 r1limg 7411 . . . . . . . . . 10  |-  ( ( x  e.  dom  R1  /\ 
Lim  x )  -> 
( R1 `  x
)  =  U_ y  e.  x  ( R1 `  y ) )
4039ancoms 441 . . . . . . . . 9  |-  ( ( Lim  x  /\  x  e.  dom  R1 )  -> 
( R1 `  x
)  =  U_ y  e.  x  ( R1 `  y ) )
41 treq 4093 . . . . . . . . 9  |-  ( ( R1 `  x )  =  U_ y  e.  x  ( R1 `  y )  ->  ( Tr  ( R1 `  x
)  <->  Tr  U_ y  e.  x  ( R1 `  y ) ) )
4240, 41syl 17 . . . . . . . 8  |-  ( ( Lim  x  /\  x  e.  dom  R1 )  -> 
( Tr  ( R1
`  x )  <->  Tr  U_ y  e.  x  ( R1 `  y ) ) )
4338, 42syl5ibr 214 . . . . . . 7  |-  ( ( Lim  x  /\  x  e.  dom  R1 )  -> 
( A. y  e.  x  Tr  ( R1
`  y )  ->  Tr  ( R1 `  x
) ) )
4443impancom 429 . . . . . 6  |-  ( ( Lim  x  /\  A. y  e.  x  Tr  ( R1 `  y ) )  ->  ( x  e.  dom  R1  ->  Tr  ( R1 `  x ) ) )
45 ndmfv 5486 . . . . . . . 8  |-  ( -.  x  e.  dom  R1  ->  ( R1 `  x
)  =  (/) )
4645, 10syl 17 . . . . . . 7  |-  ( -.  x  e.  dom  R1  ->  ( Tr  ( R1
`  x )  <->  Tr  (/) ) )
4721, 46mpbiri 226 . . . . . 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 425 . . . 4  |-  ( Lim  x  ->  ( A. y  e.  x  Tr  ( R1 `  y )  ->  Tr  ( R1 `  x ) ) )
5011, 14, 17, 20, 21, 37, 49tfinds 4622 . . 3  |-  ( A  e.  On  ->  Tr  ( R1 `  A ) )
516, 50syl 17 . 2  |-  ( A  e.  dom  R1  ->  Tr  ( R1 `  A
) )
52 ndmfv 5486 . . . 4  |-  ( -.  A  e.  dom  R1  ->  ( R1 `  A
)  =  (/) )
53 treq 4093 . . . 4  |-  ( ( R1 `  A )  =  (/)  ->  ( Tr  ( R1 `  A
)  <->  Tr  (/) ) )
5452, 53syl 17 . . 3  |-  ( -.  A  e.  dom  R1  ->  ( Tr  ( R1
`  A )  <->  Tr  (/) ) )
5521, 54mpbiri 226 . 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 5    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518    C_ wss 3127   (/)c0 3430   ~Pcpw 3599   U_ciun 3879   Tr wtr 4087   Ord word 4363   Oncon0 4364   Lim wlim 4365   suc csuc 4366   dom cdm 4661   Fun wfun 4667   ` cfv 4673   R1cr1 7402
This theorem is referenced by:  r1tr2  7417  r1ordg  7418  r1ord3g  7419  r1ord2  7421  r1sssuc  7423  r1pwss  7424  r1val1  7426  rankwflemb  7433  r1elwf  7436  r1elssi  7445  uniwf  7459  tcrank  7522  ackbij2lem3  7835  r1limwun  8326  tskr1om2  8358
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-recs 6356  df-rdg 6391  df-r1 7404
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