Theorem r1tr 5765

Description: The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202.

Assertion

Ref	Expression
r1tr	|- Tr (R1` A)

Proof of Theorem r1tr

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . 4 |- (x = (/) -> (R1` x) = (R1` (/)))
2		treq 3417	. . . 4 |- ((R1` x) = (R1` (/)) -> (Tr (R1` x) <-> Tr (R1` (/))))
3	1, 2	syl 12	. . 3 |- (x = (/) -> (Tr (R1` x) <-> Tr (R1` (/))))
4		fveq2 4681	. . . 4 |- (x = y -> (R1` x) = (R1` y))
5		treq 3417	. . . 4 |- ((R1` x) = (R1` y) -> (Tr (R1` x) <-> Tr (R1` y)))
6	4, 5	syl 12	. . 3 |- (x = y -> (Tr (R1` x) <-> Tr (R1` y)))
7		fveq2 4681	. . . 4 |- (x = suc y -> (R1` x) = (R1` suc y))
8		treq 3417	. . . 4 |- ((R1` x) = (R1` suc y) -> (Tr (R1` x) <-> Tr (R1` suc y)))
9	7, 8	syl 12	. . 3 |- (x = suc y -> (Tr (R1` x) <-> Tr (R1` suc y)))
10		fveq2 4681	. . . 4 |- (x = A -> (R1` x) = (R1` A))
11		treq 3417	. . . 4 |- ((R1` x) = (R1` A) -> (Tr (R1` x) <-> Tr (R1` A)))
12	10, 11	syl 12	. . 3 |- (x = A -> (Tr (R1` x) <-> Tr (R1` A)))
13		tr0 3423	. . . 4 |- Tr (/)
14		r10 5762	. . . . 5 |- (R1` (/)) = (/)
15		treq 3417	. . . . 5 |- ((R1` (/)) = (/) -> (Tr (R1` (/)) <-> Tr (/)))
16	14, 15	ax-mp 7	. . . 4 |- (Tr (R1` (/)) <-> Tr (/))
17	13, 16	mpbir 207	. . 3 |- Tr (R1` (/))
18		r1suc 5763	. . . . . . . . . 10 |- (y e. On -> (R1` suc y) = ~P(R1` y))
19	18	eleq2d 1964	. . . . . . . . 9 |- (y e. On -> (x e. (R1` suc y) <-> x e. ~P(R1` y)))
20		visset 2295	. . . . . . . . . 10 |- x e. _V
21	20	elpw 3037	. . . . . . . . 9 |- (x e. ~P(R1` y) <-> x C_ (R1` y))
22	19, 21	syl6bb 595	. . . . . . . 8 |- (y e. On -> (x e. (R1` suc y) <-> x C_ (R1` y)))
23	22	adantr 425	. . . . . . 7 |- ((y e. On /\ Tr (R1` y)) -> (x e. (R1` suc y) <-> x C_ (R1` y)))
24		ssel 2615	. . . . . . . . . 10 |- (x C_ (R1` y) -> (z e. x -> z e. (R1` y)))
25		dftr4 3416	. . . . . . . . . . . 12 |- (Tr (R1` y) <-> (R1` y) C_ ~P(R1` y))
26		ssel 2615	. . . . . . . . . . . 12 |- ((R1` y) C_ ~P(R1` y) -> (z e. (R1` y) -> z e. ~P(R1` y)))
27	25, 26	sylbi 216	. . . . . . . . . . 11 |- (Tr (R1` y) -> (z e. (R1` y) -> z e. ~P(R1` y)))
28	18	eleq2d 1964	. . . . . . . . . . . 12 |- (y e. On -> (z e. (R1` suc y) <-> z e. ~P(R1` y)))
29	28	biimprd 171	. . . . . . . . . . 11 |- (y e. On -> (z e. ~P(R1` y) -> z e. (R1` suc y)))
30	27, 29	sylan9r 519	. . . . . . . . . 10 |- ((y e. On /\ Tr (R1` y)) -> (z e. (R1` y) -> z e. (R1` suc y)))
31	24, 30	sylan9r 519	. . . . . . . . 9 |- (((y e. On /\ Tr (R1` y)) /\ x C_ (R1` y)) -> (z e. x -> z e. (R1` suc y)))
32	31	ssrdv 2622	. . . . . . . 8 |- (((y e. On /\ Tr (R1` y)) /\ x C_ (R1` y)) -> x C_ (R1` suc y))
33	32	ex 402	. . . . . . 7 |- ((y e. On /\ Tr (R1` y)) -> (x C_ (R1` y) -> x C_ (R1` suc y)))
34	23, 33	sylbid 220	. . . . . 6 |- ((y e. On /\ Tr (R1` y)) -> (x e. (R1` suc y) -> x C_ (R1` suc y)))
35	34	r19.21aiv 2175	. . . . 5 |- ((y e. On /\ Tr (R1` y)) -> A.x e. (R1` suc y)x C_ (R1` suc y))
36		dftr3 3415	. . . . 5 |- (Tr (R1` suc y) <-> A.x e. (R1` suc y)x C_ (R1` suc y))
37	35, 36	sylibr 217	. . . 4 |- ((y e. On /\ Tr (R1` y)) -> Tr (R1` suc y))
38	37	ex 402	. . 3 |- (y e. On -> (Tr (R1` y) -> Tr (R1` suc y)))
39		r1lim 5764	. . . . . . . . . . 11 |- ((x e. _V /\ Lim x) -> (R1` x) = U_y e. x (R1` y))
40	20, 39	mpan 759	. . . . . . . . . 10 |- (Lim x -> (R1` x) = U_y e. x (R1` y))
41	40	eleq2d 1964	. . . . . . . . 9 |- (Lim x -> (z e. (R1` x) <-> z e. U_y e. x (R1` y)))
42		eliun 3259	. . . . . . . . . 10 |- (z e. U_y e. x (R1` y) <-> E.y e. x z e. (R1` y))
43	42	biimpi 168	. . . . . . . . 9 |- (z e. U_y e. x (R1` y) -> E.y e. x z e. (R1` y))
44	41, 43	syl6bi 231	. . . . . . . 8 |- (Lim x -> (z e. (R1` x) -> E.y e. x z e. (R1` y)))
45		hbra1 2147	. . . . . . . . 9 |- (A.y e. x Tr (R1` y) -> A.yA.y e. x Tr (R1` y))
46		ra4 2155	. . . . . . . . . 10 |- (A.y e. x Tr (R1` y) -> (y e. x -> Tr (R1` y)))
47		trss 3421	. . . . . . . . . 10 |- (Tr (R1` y) -> (z e. (R1` y) -> z C_ (R1` y)))
48	46, 47	syl6 25	. . . . . . . . 9 |- (A.y e. x Tr (R1` y) -> (y e. x -> (z e. (R1` y) -> z C_ (R1` y))))
49	45, 48	reximdai 2199	. . . . . . . 8 |- (A.y e. x Tr (R1` y) -> (E.y e. x z e. (R1` y) -> E.y e. x z C_ (R1` y)))
50	44, 49	sylan9 517	. . . . . . 7 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> (z e. (R1` x) -> E.y e. x z C_ (R1` y)))
51	40	sseq2d 2645	. . . . . . . . 9 |- (Lim x -> (z C_ (R1` x) <-> z C_ U_y e. x (R1` y)))
52		ssiun 3293	. . . . . . . . 9 |- (E.y e. x z C_ (R1` y) -> z C_ U_y e. x (R1` y))
53	51, 52	syl5bir 227	. . . . . . . 8 |- (Lim x -> (E.y e. x z C_ (R1` y) -> z C_ (R1` x)))
54	53	adantr 425	. . . . . . 7 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> (E.y e. x z C_ (R1` y) -> z C_ (R1` x)))
55	50, 54	syld 30	. . . . . 6 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> (z e. (R1` x) -> z C_ (R1` x)))
56	55	r19.21aiv 2175	. . . . 5 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> A.z e. (R1` x)z C_ (R1` x))
57		dftr3 3415	. . . . 5 |- (Tr (R1` x) <-> A.z e. (R1` x)z C_ (R1` x))
58	56, 57	sylibr 217	. . . 4 |- ((Lim x /\ A.y e. x Tr (R1` y)) -> Tr (R1` x))
59	58	ex 402	. . 3 |- (Lim x -> (A.y e. x Tr (R1` y) -> Tr (R1` x)))
60	3, 6, 9, 12, 17, 38, 59	tfinds 3942	. 2 |- (A e. On -> Tr (R1` A))
61		r1fnon 5761	. . . . . . . 8 |- R1 Fn On
62		fndm 4512	. . . . . . . 8 |- (R1 Fn On -> dom R1 = On)
63	61, 62	ax-mp 7	. . . . . . 7 |- dom R1 = On
64	63	eleq2i 1961	. . . . . 6 |- (A e. dom R1 <-> A e. On)
65	64	notbii 204	. . . . 5 |- (-. A e. dom R1 <-> -. A e. On)
66		ndmfv 4702	. . . . 5 |- (-. A e. dom R1 -> (R1` A) = (/))
67	65, 66	sylbir 218	. . . 4 |- (-. A e. On -> (R1` A) = (/))
68		treq 3417	. . . 4 |- ((R1` A) = (/) -> (Tr (R1` A) <-> Tr (/)))
69	67, 68	syl 12	. . 3 |- (-. A e. On -> (Tr (R1` A) <-> Tr (/)))
70	13, 69	mpbiri 211	. 2 |- (-. A e. On -> Tr (R1` A))
71	60, 70	pm2.61i 140	1 |- Tr (R1` A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  U_ciun 3255  Tr wtr 3411  Oncon0 3657  Lim wlim 3658  suc csuc 3659  dom cdm 3986   Fn wfn 3993  ` cfv 3998  R1cr1 5748

This theorem is referenced by:  r1ord 5766  r1ord2 5767  r1subsuc 14439  uncum 14440

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750

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