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.

Step	Hyp	Ref	Expression
1		r1funlim 8085	. . . . . 6 |- ( Fun R1 /\ Lim dom R1 )
2	1	simpri 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 )
5	2, 3, 4	mp2b 10	. . . 4 |- dom R1 C_ On
6	5	sseli 3461	. . 3 |- ( A e. dom R1 -> A e. On )
7		fveq2 5800	. . . . . 6 |- ( x = (/) -> ( R1 ` x ) = ( R1 ` (/) ) )
8		r10 8087	. . . . . 6 |- ( R1 ` (/) ) = (/)
9	7, 8	syl6eq 2511	. . . . 5 |- ( x = (/) -> ( R1 ` x ) = (/) )
10		treq 4500	. . . . 5 |- ( ( R1 ` x ) = (/) -> ( Tr ( R1 ` x ) <-> Tr (/) ) )
11	9, 10	syl 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 ) ) )
14	12, 13	syl 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 ) ) )
17	15, 16	syl 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 ) ) )
20	18, 19	syl 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 ) )
23	2, 22	ax-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 ) )
26	24, 25	sylib 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 ) ) )
29	27, 28	syl 16	. . . . . . . 8 |- ( y e. dom R1 -> ( Tr ( R1 ` suc y ) <-> Tr ~P ( R1 ` y ) ) )
30	26, 29	syl5ibrcom 222	. . . . . . 7 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> ( y e. dom R1 -> Tr ( R1 ` suc y ) ) )
31	23, 30	syl5bir 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 (/) ) )
34	32, 33	syl 16	. . . . . . 7 |- ( -. suc y e. dom R1 -> ( Tr ( R1 ` suc y ) <-> Tr (/) ) )
35	21, 34	mpbiri 233	. . . . . 6 |- ( -. suc y e. dom R1 -> Tr ( R1 ` suc y ) )
36	31, 35	pm2.61d1 159	. . . . 5 |- ( ( y e. On /\ Tr ( R1 ` y ) ) -> Tr ( R1 ` suc y ) )
37	36	ex 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 ) )
40	39	ancoms 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 ) ) )
42	40, 41	syl 16	. . . . . . . 8 |- ( ( Lim x /\ x e. dom R1 ) -> ( Tr ( R1 ` x ) <-> Tr U_ y e. x ( R1 ` y ) ) )
43	38, 42	syl5ibr 221	. . . . . . 7 |- ( ( Lim x /\ x e. dom R1 ) -> ( A. y e. x Tr ( R1 ` y ) -> Tr ( R1 ` x ) ) )
44	43	impancom 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 ) = (/) )
46	45, 10	syl 16	. . . . . . 7 |- ( -. x e. dom R1 -> ( Tr ( R1 ` x ) <-> Tr (/) ) )
47	21, 46	mpbiri 233	. . . . . 6 |- ( -. x e. dom R1 -> Tr ( R1 ` x ) )
48	44, 47	pm2.61d1 159	. . . . 5 |- ( ( Lim x /\ A. y e. x Tr ( R1 ` y ) ) -> Tr ( R1 ` x ) )
49	48	ex 434	. . . 4 |- ( Lim x -> ( A. y e. x Tr ( R1 ` y ) -> Tr ( R1 ` x ) ) )
50	11, 14, 17, 20, 21, 37, 49	tfinds 6581	. . 3 |- ( A e. On -> Tr ( R1 ` A ) )
51	6, 50	syl 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 (/) ) )
54	52, 53	syl 16	. . 3 |- ( -. A e. dom R1 -> ( Tr ( R1 ` A ) <-> Tr (/) ) )
55	21, 54	mpbiri 233	. 2 |- ( -. A e. dom R1 -> Tr ( R1 ` A ) )
56	51, 55	pm2.61i 164	1 |- Tr ( R1 ` A )

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