Theorem r1ord 5766

Description: Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77.

Assertion

Ref	Expression
r1ord	|- (B e. On -> (A e. B -> (R1` A) e. (R1` B)))

Proof of Theorem r1ord

Step	Hyp	Ref	Expression
1		simpl 346	. . . 4 |- ((B e. On /\ A e. B) -> B e. On)
2		onelon 3683	. . . . 5 |- ((B e. On /\ A e. B) -> A e. On)
3		suceloni 3894	. . . . 5 |- (A e. On -> suc A e. On)
4	2, 3	syl 12	. . . 4 |- ((B e. On /\ A e. B) -> suc A e. On)
5		eloni 3667	. . . . . 6 |- (B e. On -> Ord B)
6		ordsucss 3899	. . . . . 6 |- (Ord B -> (A e. B -> suc A C_ B))
7	5, 6	syl 12	. . . . 5 |- (B e. On -> (A e. B -> suc A C_ B))
8	7	imp 377	. . . 4 |- ((B e. On /\ A e. B) -> suc A C_ B)
9		eleq2 1958	. . . . . 6 |- (x = suc A -> (A e. x <-> A e. suc A))
10		fveq2 4681	. . . . . . 7 |- (x = suc A -> (R1` x) = (R1` suc A))
11	10	eleq2d 1964	. . . . . 6 |- (x = suc A -> ((R1` A) e. (R1` x) <-> (R1` A) e. (R1` suc A)))
12	9, 11	imbi12d 688	. . . . 5 |- (x = suc A -> ((A e. x -> (R1` A) e. (R1` x)) <-> (A e. suc A -> (R1` A) e. (R1` suc A))))
13		eleq2 1958	. . . . . 6 |- (x = y -> (A e. x <-> A e. y))
14		fveq2 4681	. . . . . . 7 |- (x = y -> (R1` x) = (R1` y))
15	14	eleq2d 1964	. . . . . 6 |- (x = y -> ((R1` A) e. (R1` x) <-> (R1` A) e. (R1` y)))
16	13, 15	imbi12d 688	. . . . 5 |- (x = y -> ((A e. x -> (R1` A) e. (R1` x)) <-> (A e. y -> (R1` A) e. (R1` y))))
17		eleq2 1958	. . . . . 6 |- (x = suc y -> (A e. x <-> A e. suc y))
18		fveq2 4681	. . . . . . 7 |- (x = suc y -> (R1` x) = (R1` suc y))
19	18	eleq2d 1964	. . . . . 6 |- (x = suc y -> ((R1` A) e. (R1` x) <-> (R1` A) e. (R1` suc y)))
20	17, 19	imbi12d 688	. . . . 5 |- (x = suc y -> ((A e. x -> (R1` A) e. (R1` x)) <-> (A e. suc y -> (R1` A) e. (R1` suc y))))
21		eleq2 1958	. . . . . 6 |- (x = B -> (A e. x <-> A e. B))
22		fveq2 4681	. . . . . . 7 |- (x = B -> (R1` x) = (R1` B))
23	22	eleq2d 1964	. . . . . 6 |- (x = B -> ((R1` A) e. (R1` x) <-> (R1` A) e. (R1` B)))
24	21, 23	imbi12d 688	. . . . 5 |- (x = B -> ((A e. x -> (R1` A) e. (R1` x)) <-> (A e. B -> (R1` A) e. (R1` B))))
25		onelon 3683	. . . . . . 7 |- ((suc A e. On /\ A e. suc A) -> A e. On)
26		r1suc 5763	. . . . . . . 8 |- (A e. On -> (R1` suc A) = ~P(R1` A))
27		fvex 4689	. . . . . . . . 9 |- (R1` A) e. _V
28	27	pwid 3042	. . . . . . . 8 |- (R1` A) e. ~P(R1` A)
29	26, 28	syl5eleqr 1978	. . . . . . 7 |- (A e. On -> (R1` A) e. (R1` suc A))
30	25, 29	syl 12	. . . . . 6 |- ((suc A e. On /\ A e. suc A) -> (R1` A) e. (R1` suc A))
31	30	ex 402	. . . . 5 |- (suc A e. On -> (A e. suc A -> (R1` A) e. (R1` suc A)))
32		r1suc 5763	. . . . . . . . . . . . . 14 |- (y e. On -> (R1` suc y) = ~P(R1` y))
33		fvex 4689	. . . . . . . . . . . . . . 15 |- (R1` y) e. _V
34	33	pwid 3042	. . . . . . . . . . . . . 14 |- (R1` y) e. ~P(R1` y)
35	32, 34	syl5eleqr 1978	. . . . . . . . . . . . 13 |- (y e. On -> (R1` y) e. (R1` suc y))
36		r1tr 5765	. . . . . . . . . . . . . 14 |- Tr (R1` suc y)
37		trss 3421	. . . . . . . . . . . . . 14 |- (Tr (R1` suc y) -> ((R1` y) e. (R1` suc y) -> (R1` y) C_ (R1` suc y)))
38	36, 37	ax-mp 7	. . . . . . . . . . . . 13 |- ((R1` y) e. (R1` suc y) -> (R1` y) C_ (R1` suc y))
39	35, 38	syl 12	. . . . . . . . . . . 12 |- (y e. On -> (R1` y) C_ (R1` suc y))
40	39	sseld 2619	. . . . . . . . . . 11 |- (y e. On -> ((R1` A) e. (R1` y) -> (R1` A) e. (R1` suc y)))
41	40	imim2d 28	. . . . . . . . . 10 |- (y e. On -> ((A e. y -> (R1` A) e. (R1` y)) -> (A e. y -> (R1` A) e. (R1` suc y))))
42		elisset 2299	. . . . . . . . . . . . 13 |- (suc A e. On -> suc A e. _V)
43		sucexb 3890	. . . . . . . . . . . . 13 |- (A e. _V <-> suc A e. _V)
44	42, 43	sylibr 217	. . . . . . . . . . . 12 |- (suc A e. On -> A e. _V)
45		sucssel 3763	. . . . . . . . . . . 12 |- (A e. _V -> (suc A C_ y -> A e. y))
46	44, 45	syl 12	. . . . . . . . . . 11 |- (suc A e. On -> (suc A C_ y -> A e. y))
47	46	imp 377	. . . . . . . . . 10 |- ((suc A e. On /\ suc A C_ y) -> A e. y)
48	41, 47	syl7 26	. . . . . . . . 9 |- (y e. On -> ((A e. y -> (R1` A) e. (R1` y)) -> ((suc A e. On /\ suc A C_ y) -> (R1` A) e. (R1` suc y))))
49	48	a1d 15	. . . . . . . 8 |- (y e. On -> (A e. suc y -> ((A e. y -> (R1` A) e. (R1` y)) -> ((suc A e. On /\ suc A C_ y) -> (R1` A) e. (R1` suc y)))))
50	49	com24 41	. . . . . . 7 |- (y e. On -> ((suc A e. On /\ suc A C_ y) -> ((A e. y -> (R1` A) e. (R1` y)) -> (A e. suc y -> (R1` A) e. (R1` suc y)))))
51	50	exp3a 405	. . . . . 6 |- (y e. On -> (suc A e. On -> (suc A C_ y -> ((A e. y -> (R1` A) e. (R1` y)) -> (A e. suc y -> (R1` A) e. (R1` suc y))))))
52	51	imp31 389	. . . . 5 |- (((y e. On /\ suc A e. On) /\ suc A C_ y) -> ((A e. y -> (R1` A) e. (R1` y)) -> (A e. suc y -> (R1` A) e. (R1` suc y))))
53		limsuc 3933	. . . . . . . . . . . 12 |- (Lim x -> (A e. x <-> suc A e. x))
54	53	biimpa 460	. . . . . . . . . . 11 |- ((Lim x /\ A e. x) -> suc A e. x)
55		onelon 3683	. . . . . . . . . . . . 13 |- ((x e. On /\ A e. x) -> A e. On)
56		limord 3723	. . . . . . . . . . . . . 14 |- (Lim x -> Ord x)
57		visset 2295	. . . . . . . . . . . . . . 15 |- x e. _V
58	57	elon 3666	. . . . . . . . . . . . . 14 |- (x e. On <-> Ord x)
59	56, 58	sylibr 217	. . . . . . . . . . . . 13 |- (Lim x -> x e. On)
60	55, 59	sylan 497	. . . . . . . . . . . 12 |- ((Lim x /\ A e. x) -> A e. On)
61	60, 29	syl 12	. . . . . . . . . . 11 |- ((Lim x /\ A e. x) -> (R1` A) e. (R1` suc A))
62		fveq2 4681	. . . . . . . . . . . . 13 |- (y = suc A -> (R1` y) = (R1` suc A))
63	62	eleq2d 1964	. . . . . . . . . . . 12 |- (y = suc A -> ((R1` A) e. (R1` y) <-> (R1` A) e. (R1` suc A)))
64	63	rcla4ev 2381	. . . . . . . . . . 11 |- ((suc A e. x /\ (R1` A) e. (R1` suc A)) -> E.y e. x (R1` A) e. (R1` y))
65	54, 61, 64	syl11anc 524	. . . . . . . . . 10 |- ((Lim x /\ A e. x) -> E.y e. x (R1` A) e. (R1` y))
66		eliun 3259	. . . . . . . . . 10 |- ((R1` A) e. U_y e. x (R1` y) <-> E.y e. x (R1` A) e. (R1` y))
67	65, 66	sylibr 217	. . . . . . . . 9 |- ((Lim x /\ A e. x) -> (R1` A) e. U_y e. x (R1` y))
68		r1lim 5764	. . . . . . . . . . . 12 |- ((x e. _V /\ Lim x) -> (R1` x) = U_y e. x (R1` y))
69	57, 68	mpan 759	. . . . . . . . . . 11 |- (Lim x -> (R1` x) = U_y e. x (R1` y))
70	69	eleq2d 1964	. . . . . . . . . 10 |- (Lim x -> ((R1` A) e. (R1` x) <-> (R1` A) e. U_y e. x (R1` y)))
71	70	adantr 425	. . . . . . . . 9 |- ((Lim x /\ A e. x) -> ((R1` A) e. (R1` x) <-> (R1` A) e. U_y e. x (R1` y)))
72	67, 71	mpbird 213	. . . . . . . 8 |- ((Lim x /\ A e. x) -> (R1` A) e. (R1` x))
73	72	ex 402	. . . . . . 7 |- (Lim x -> (A e. x -> (R1` A) e. (R1` x)))
74	73	ad2antrr 440	. . . . . 6 |- (((Lim x /\ suc A e. On) /\ suc A C_ x) -> (A e. x -> (R1` A) e. (R1` x)))
75	74	a1d 15	. . . . 5 |- (((Lim x /\ suc A e. On) /\ suc A C_ x) -> (A.y e. x (suc A C_ y -> (A e. y -> (R1` A) e. (R1` y))) -> (A e. x -> (R1` A) e. (R1` x))))
76	12, 16, 20, 24, 31, 52, 75	tfindsg 3944	. . . 4 |- (((B e. On /\ suc A e. On) /\ suc A C_ B) -> (A e. B -> (R1` A) e. (R1` B)))
77	1, 4, 8, 76	syl21anc 1099	. . 3 |- ((B e. On /\ A e. B) -> (A e. B -> (R1` A) e. (R1` B)))
78	77	ex 402	. 2 |- (B e. On -> (A e. B -> (A e. B -> (R1` A) e. (R1` B))))
79	78	pm2.43d 79	1 |- (B e. On -> (A e. B -> (R1` A) e. (R1` B)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032  U_ciun 3255  Tr wtr 3411  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659  ` cfv 3998  R1cr1 5748

This theorem is referenced by:  r1ord2 5767

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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