Theorem r1tr 3498

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 2832	. . . 4 ⊢ (x = ∅ → (R1 ‘x) = (R1 ‘∅))
2		treq 2047	. . . 4 ⊢ ((R1 ‘x) = (R1 ‘∅) → (Tr (R1 ‘x) ↔ Tr (R1 ‘∅)))
3	1, 2	syl 12	. . 3 ⊢ (x = ∅ → (Tr (R1 ‘x) ↔ Tr (R1 ‘∅)))
4		fveq2 2832	. . . 4 ⊢ (x = y → (R1 ‘x) = (R1 ‘y))
5		treq 2047	. . . 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 2832	. . . 4 ⊢ (x = suc y → (R1 ‘x) = (R1 ‘suc y))
8		treq 2047	. . . 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 2832	. . . 4 ⊢ (x = A → (R1 ‘x) = (R1 ‘A))
11		treq 2047	. . . 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 2052	. . . 4 ⊢ Tr ∅
14		r10 3495	. . . . 5 ⊢ (R1 ‘∅) = ∅
15		treq 2047	. . . . 5 ⊢ ((R1 ‘∅) = ∅ → (Tr (R1 ‘∅) ↔ Tr ∅))
16	14, 15	ax-mp 6	. . . 4 ⊢ (Tr (R1 ‘∅) ↔ Tr ∅)
17	13, 16	mpbir 165	. . 3 ⊢ Tr (R1 ‘∅)
18		r1suc 3496	. . . . . . . . . 10 ⊢ (y ∈ On → (R1 ‘suc y) = ℘(R1 ‘y))
19	18	eleq2d 1156	. . . . . . . . 9 ⊢ (y ∈ On → (x ∈ (R1 ‘suc y) ↔ x ∈ ℘(R1 ‘y)))
20		visset 1350	. . . . . . . . . 10 ⊢ x ∈ V
21	20	elpw 1801	. . . . . . . . 9 ⊢ (x ∈ ℘(R1 ‘y) ↔ x ⊆ (R1 ‘y))
22	19, 21	syl6bb 414	. . . . . . . 8 ⊢ (y ∈ On → (x ∈ (R1 ‘suc y) ↔ x ⊆ (R1 ‘y)))
23	22	adantr 306	. . . . . . 7 ⊢ ((y ∈ On ∧ Tr (R1 ‘y)) → (x ∈ (R1 ‘suc y) ↔ x ⊆ (R1 ‘y)))
24		ssel 1502	. . . . . . . . . 10 ⊢ (x ⊆ (R1 ‘y) → (z ∈ x → z ∈ (R1 ‘y)))
25		dftr4 2046	. . . . . . . . . . . 12 ⊢ (Tr (R1 ‘y) ↔ (R1 ‘y) ⊆ ℘(R1 ‘y))
26		ssel 1502	. . . . . . . . . . . 12 ⊢ ((R1 ‘y) ⊆ ℘(R1 ‘y) → (z ∈ (R1 ‘y) → z ∈ ℘(R1 ‘y)))
27	25, 26	sylbi 174	. . . . . . . . . . 11 ⊢ (Tr (R1 ‘y) → (z ∈ (R1 ‘y) → z ∈ ℘(R1 ‘y)))
28	18	eleq2d 1156	. . . . . . . . . . . 12 ⊢ (y ∈ On → (z ∈ (R1 ‘suc y) ↔ z ∈ ℘(R1 ‘y)))
29	28	biimprd 136	. . . . . . . . . . 11 ⊢ (y ∈ On → (z ∈ ℘(R1 ‘y) → z ∈ (R1 ‘suc y)))
30	27, 29	sylan9r 360	. . . . . . . . . 10 ⊢ ((y ∈ On ∧ Tr (R1 ‘y)) → (z ∈ (R1 ‘y) → z ∈ (R1 ‘suc y)))
31	24, 30	sylan9r 360	. . . . . . . . 9 ⊢ (((y ∈ On ∧ Tr (R1 ‘y)) ∧ x ⊆ (R1 ‘y)) → (z ∈ x → z ∈ (R1 ‘suc y)))
32	31	ssrdv 1509	. . . . . . . 8 ⊢ (((y ∈ On ∧ Tr (R1 ‘y)) ∧ x ⊆ (R1 ‘y)) → x ⊆ (R1 ‘suc y))
33	32	exp 291	. . . . . . 7 ⊢ ((y ∈ On ∧ Tr (R1 ‘y)) → (x ⊆ (R1 ‘y) → x ⊆ (R1 ‘suc y)))
34	23, 33	sylbid 178	. . . . . 6 ⊢ ((y ∈ On ∧ Tr (R1 ‘y)) → (x ∈ (R1 ‘suc y) → x ⊆ (R1 ‘suc y)))
35	34	r19.21aiv 1259	. . . . 5 ⊢ ((y ∈ On ∧ Tr (R1 ‘y)) → ∀x ∈ (R1 ‘suc y)x ⊆ (R1 ‘suc y))
36		dftr3 2045	. . . . 5 ⊢ (Tr (R1 ‘suc y) ↔ ∀x ∈ (R1 ‘suc y)x ⊆ (R1 ‘suc y))
37	35, 36	sylibr 175	. . . 4 ⊢ ((y ∈ On ∧ Tr (R1 ‘y)) → Tr (R1 ‘suc y))
38	37	exp 291	. . 3 ⊢ (y ∈ On → (Tr (R1 ‘y) → Tr (R1 ‘suc y)))
39		r1lim 3497	. . . . . . . . . . 11 ⊢ ((x ∈ V ∧ Lim x) → (R1 ‘x) = ∪y ∈ x (R1 ‘y))
40	20, 39	mpan 518	. . . . . . . . . 10 ⊢ (Lim x → (R1 ‘x) = ∪y ∈ x (R1 ‘y))
41	40	eleq2d 1156	. . . . . . . . 9 ⊢ (Lim x → (z ∈ (R1 ‘x) ↔ z ∈ ∪y ∈ x (R1 ‘y)))
42		eliun 1998	. . . . . . . . . 10 ⊢ (z ∈ ∪y ∈ x (R1 ‘y) ↔ ∃y ∈ x z ∈ (R1 ‘y))
43	42	biimp 133	. . . . . . . . 9 ⊢ (z ∈ ∪y ∈ x (R1 ‘y) → ∃y ∈ x z ∈ (R1 ‘y))
44	41, 43	syl6bi 187	. . . . . . . 8 ⊢ (Lim x → (z ∈ (R1 ‘x) → ∃y ∈ x z ∈ (R1 ‘y)))
45		hbra1 1237	. . . . . . . . 9 ⊢ (∀y ∈ x Tr (R1 ‘y) → ∀y∀y ∈ x Tr (R1 ‘y))
46		ra4 1243	. . . . . . . . . 10 ⊢ (∀y ∈ x Tr (R1 ‘y) → (y ∈ x → Tr (R1 ‘y)))
47		trss 2050	. . . . . . . . . 10 ⊢ (Tr (R1 ‘y) → (z ∈ (R1 ‘y) → z ⊆ (R1 ‘y)))
48	46, 47	syl6 23	. . . . . . . . 9 ⊢ (∀y ∈ x Tr (R1 ‘y) → (y ∈ x → (z ∈ (R1 ‘y) → z ⊆ (R1 ‘y))))
49	45, 48	r19.22d 1276	. . . . . . . 8 ⊢ (∀y ∈ x Tr (R1 ‘y) → (∃y ∈ x z ∈ (R1 ‘y) → ∃y ∈ x z ⊆ (R1 ‘y)))
50	44, 49	sylan9 359	. . . . . . 7 ⊢ ((Lim x ∧ ∀y ∈ x Tr (R1 ‘y)) → (z ∈ (R1 ‘x) → ∃y ∈ x z ⊆ (R1 ‘y)))
51	40	sseq2d 1528	. . . . . . . . 9 ⊢ (Lim x → (z ⊆ (R1 ‘x) ↔ z ⊆ ∪y ∈ x (R1 ‘y)))
52		ssiun 2018	. . . . . . . . 9 ⊢ (∃y ∈ x z ⊆ (R1 ‘y) → z ⊆ ∪y ∈ x (R1 ‘y))
53	51, 52	syl5bir 184	. . . . . . . 8 ⊢ (Lim x → (∃y ∈ x z ⊆ (R1 ‘y) → z ⊆ (R1 ‘x)))
54	53	adantr 306	. . . . . . 7 ⊢ ((Lim x ∧ ∀y ∈ x Tr (R1 ‘y)) → (∃y ∈ x z ⊆ (R1 ‘y) → z ⊆ (R1 ‘x)))
55	50, 54	syld 27	. . . . . 6 ⊢ ((Lim x ∧ ∀y ∈ x Tr (R1 ‘y)) → (z ∈ (R1 ‘x) → z ⊆ (R1 ‘x)))
56	55	r19.21aiv 1259	. . . . 5 ⊢ ((Lim x ∧ ∀y ∈ x Tr (R1 ‘y)) → ∀z ∈ (R1 ‘x)z ⊆ (R1 ‘x))
57		dftr3 2045	. . . . 5 ⊢ (Tr (R1 ‘x) ↔ ∀z ∈ (R1 ‘x)z ⊆ (R1 ‘x))
58	56, 57	sylibr 175	. . . 4 ⊢ ((Lim x ∧ ∀y ∈ x Tr (R1 ‘y)) → Tr (R1 ‘x))
59	58	exp 291	. . 3 ⊢ (Lim x → (∀y ∈ x Tr (R1 ‘y) → Tr (R1 ‘x)))
60	3, 6, 9, 12, 17, 38, 59	tfinds 2401	. 2 ⊢ (A ∈ On → Tr (R1 ‘A))
61		r1fnon 3494	. . . . . . . 8 ⊢ R1 Fn On
62		fndm 2723	. . . . . . . 8 ⊢ (R1 Fn On → dom R1 = On)
63	61, 62	ax-mp 6	. . . . . . 7 ⊢ dom R1 = On
64	63	eleq2i 1153	. . . . . 6 ⊢ (A ∈ dom R1 ↔ A ∈ On)
65	64	negbii 162	. . . . 5 ⊢ (¬ A ∈ dom R1 ↔ ¬ A ∈ On)
66		ndmfv 2848	. . . . 5 ⊢ (¬ A ∈ dom R1 → (R1 ‘A) = ∅)
67	65, 66	sylbir 176	. . . 4 ⊢ (¬ A ∈ On → (R1 ‘A) = ∅)
68		treq 2047	. . . 4 ⊢ ((R1 ‘A) = ∅ → (Tr (R1 ‘A) ↔ Tr ∅))
69	67, 68	syl 12	. . 3 ⊢ (¬ A ∈ On → (Tr (R1 ‘A) ↔ Tr ∅))
70	13, 69	mpbiri 169	. 2 ⊢ (¬ A ∈ On → Tr (R1 ‘A))
71	60, 70	pm2.61i 110	1 ⊢ Tr (R1 ‘A)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  ℘cpw 1798  ∪ciun 1994  Tr wtr 2041  Oncon0 2199  Lim wlim 2200  suc csuc 2201  dom cdm 2410   Fn wfn 2417   ‘cfv 2422  R₁cr1 3485

This theorem is referenced by:  r1ord 3499  r1ord2 3500

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-r1 3487

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