Theorem grur1 10775

Description: A characterization of Grothendieck universes, part 2. (Contributed by Mario Carneiro, 24-Jun-2013.)

Hypothesis

Ref	Expression
gruina.1	⊢ 𝐴 = (𝑈 ∩ On)

Assertion

Ref	Expression
grur1	⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → 𝑈 = (𝑅1‘𝐴))

Proof of Theorem grur1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nss 4000	. . . . 5 ⊢ (¬ 𝑈 ⊆ (𝑅1‘𝐴) ↔ ∃𝑥(𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴)))
2		fveqeq2 6872	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((rank‘𝑦) = 𝐴 ↔ (rank‘𝑥) = 𝐴))
3	2	rspcev 3581	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑈 ∧ (rank‘𝑥) = 𝐴) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴)
4	3	ex 416	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → ((rank‘𝑥) = 𝐴 → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
5	4	ad2antrl 738	. . . . . . . 8 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ((rank‘𝑥) = 𝐴 → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
6		simplr 778	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → 𝑈 ∈ ∪ (𝑅1 “ On))
7		simprl 780	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → 𝑥 ∈ 𝑈)
8		r1elssi 9760	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ ∪ (𝑅1 “ On) → 𝑈 ⊆ ∪ (𝑅1 “ On))
9	8	sseld 3935	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝑈 → 𝑥 ∈ ∪ (𝑅1 “ On)))
10	6, 7, 9	sylc 65	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → 𝑥 ∈ ∪ (𝑅1 “ On))
11		tcrank 9839	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (rank‘𝑥) = (rank “ (TC‘𝑥)))
12	10, 11	syl 17	. . . . . . . . . 10 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → (rank‘𝑥) = (rank “ (TC‘𝑥)))
13	12	eleq2d 2847	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → (𝐴 ∈ (rank‘𝑥) ↔ 𝐴 ∈ (rank “ (TC‘𝑥))))
14		gruelss 10749	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ 𝑈)
15		grutr 10748	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ Univ → Tr 𝑈)
16	15	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → Tr 𝑈)
17		vex 3457	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
18		tcmin 9691	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ V → ((𝑥 ⊆ 𝑈 ∧ Tr 𝑈) → (TC‘𝑥) ⊆ 𝑈))
19	17, 18	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝑈 ∧ Tr 𝑈) → (TC‘𝑥) ⊆ 𝑈)
20	14, 16, 19	syl2anc 593	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → (TC‘𝑥) ⊆ 𝑈)
21		rankf 9749	. . . . . . . . . . . . 13 ⊢ rank:∪ (𝑅1 “ On)⟶On
22		ffun 6690	. . . . . . . . . . . . 13 ⊢ (rank:∪ (𝑅1 “ On)⟶On → Fun rank)
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun rank
24		fvelima 6928	. . . . . . . . . . . 12 ⊢ ((Fun rank ∧ 𝐴 ∈ (rank “ (TC‘𝑥))) → ∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴)
25	23, 24	mpan 700	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴)
26		ssrexv 4006	. . . . . . . . . . 11 ⊢ ((TC‘𝑥) ⊆ 𝑈 → (∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴 → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
27	20, 25, 26	syl2im 40	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
28	27	ad2ant2r 757	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
29	13, 28	sylbid 242	. . . . . . . 8 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → (𝐴 ∈ (rank‘𝑥) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
30		simprr 782	. . . . . . . . . 10 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ¬ 𝑥 ∈ (𝑅1‘𝐴))
31		ne0i 4293	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑈 → 𝑈 ≠ ∅)
32		gruina.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = (𝑈 ∩ On)
33	32	gruina 10773	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝐴 ∈ Inacc)
34	31, 33	sylan2 602	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ Inacc)
35		inawina 10645	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
36		winaon 10643	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ On)
37	34, 35, 36	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ On)
38		r1fnon 9722	. . . . . . . . . . . . . 14 ⊢ 𝑅1 Fn On
39		fndm 6620	. . . . . . . . . . . . . 14 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
40	38, 39	ax-mp 5	. . . . . . . . . . . . 13 ⊢ dom 𝑅1 = On
41	37, 40	eleqtrrdi 2872	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ dom 𝑅1)
42	41	ad2ant2r 757	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → 𝐴 ∈ dom 𝑅1)
43		rankr1ag 9757	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
44	10, 42, 43	syl2anc 593	. . . . . . . . . 10 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
45	30, 44	mtbid 326	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ¬ (rank‘𝑥) ∈ 𝐴)
46		rankon 9750	. . . . . . . . . . . . 13 ⊢ (rank‘𝑥) ∈ On
47		eloni 6352	. . . . . . . . . . . . . 14 ⊢ ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
48		eloni 6352	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → Ord 𝐴)
49		ordtri3or 6374	. . . . . . . . . . . . . 14 ⊢ ((Ord (rank‘𝑥) ∧ Ord 𝐴) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
50	47, 48, 49	syl2an 605	. . . . . . . . . . . . 13 ⊢ (((rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
51	46, 37, 50	sylancr 596	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
52		3orass 1100	. . . . . . . . . . . 12 ⊢ (((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)) ↔ ((rank‘𝑥) ∈ 𝐴 ∨ ((rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥))))
53	51, 52	sylib 220	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → ((rank‘𝑥) ∈ 𝐴 ∨ ((rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥))))
54	53	ord 875	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → (¬ (rank‘𝑥) ∈ 𝐴 → ((rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥))))
55	54	ad2ant2r 757	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → (¬ (rank‘𝑥) ∈ 𝐴 → ((rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥))))
56	45, 55	mpd 15	. . . . . . . 8 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ((rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
57	5, 29, 56	mpjaod 871	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴)
58	57	ex 416	. . . . . 6 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → ((𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴)) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
59	58	exlimdv 1952	. . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → (∃𝑥(𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴)) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
60	1, 59	biimtrid 244	. . . 4 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → (¬ 𝑈 ⊆ (𝑅1‘𝐴) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
61		simpll 776	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ∈ Univ)
62		ne0i 4293	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑈 → 𝑈 ≠ ∅)
63	62, 33	sylan2 602	. . . . . . . . 9 ⊢ ((𝑈 ∈ Univ ∧ 𝑦 ∈ 𝑈) → 𝐴 ∈ Inacc)
64	63	ad2ant2r 757	. . . . . . . 8 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ∈ Inacc)
65	64, 35, 36	3syl 18	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ∈ On)
66		simprl 780	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑦 ∈ 𝑈)
67		fveq2 6863	. . . . . . . . . 10 ⊢ ((rank‘𝑦) = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘𝐴))
68	67	ad2antll 739	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘(rank‘𝑦)) = (cf‘𝐴))
69		elina 10642	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
70	69	simp2bi 1158	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
71	64, 70	syl 17	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘𝐴) = 𝐴)
72	68, 71	eqtrd 2796	. . . . . . . 8 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘(rank‘𝑦)) = 𝐴)
73		rankcf 10732	. . . . . . . . 9 ⊢ ¬ 𝑦 ≺ (cf‘(rank‘𝑦))
74		fvex 6876	. . . . . . . . . 10 ⊢ (cf‘(rank‘𝑦)) ∈ V
75		vex 3457	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
76		domtri 10510	. . . . . . . . . 10 ⊢ (((cf‘(rank‘𝑦)) ∈ V ∧ 𝑦 ∈ V) → ((cf‘(rank‘𝑦)) ≼ 𝑦 ↔ ¬ 𝑦 ≺ (cf‘(rank‘𝑦))))
77	74, 75, 76	mp2an 702	. . . . . . . . 9 ⊢ ((cf‘(rank‘𝑦)) ≼ 𝑦 ↔ ¬ 𝑦 ≺ (cf‘(rank‘𝑦)))
78	73, 77	mpbir 233	. . . . . . . 8 ⊢ (cf‘(rank‘𝑦)) ≼ 𝑦
79	72, 78	eqbrtrrdi 5139	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ≼ 𝑦)
80		grudomon 10772	. . . . . . 7 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝑦 ∈ 𝑈 ∧ 𝐴 ≼ 𝑦)) → 𝐴 ∈ 𝑈)
81	61, 65, 66, 79, 80	syl112anc 1392	. . . . . 6 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ∈ 𝑈)
82		elin 3920	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑈 ∩ On) ↔ (𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On))
83	82	biimpri 230	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On) → 𝐴 ∈ (𝑈 ∩ On))
84	83, 32	eleqtrrdi 2872	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On) → 𝐴 ∈ 𝐴)
85		ordirr 6360	. . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
86	48, 85	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ 𝐴 ∈ 𝐴)
87	86	adantl 485	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On) → ¬ 𝐴 ∈ 𝐴)
88	84, 87	pm2.21dd 197	. . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On) → 𝑈 ⊆ (𝑅1‘𝐴))
89	81, 65, 88	syl2anc 593	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ⊆ (𝑅1‘𝐴))
90	89	rexlimdvaa 3163	. . . 4 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → (∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴 → 𝑈 ⊆ (𝑅1‘𝐴)))
91	60, 90	syld 47	. . 3 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → (¬ 𝑈 ⊆ (𝑅1‘𝐴) → 𝑈 ⊆ (𝑅1‘𝐴)))
92	91	pm2.18d 127	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → 𝑈 ⊆ (𝑅1‘𝐴))
93	32	grur1a 10774	. . 3 ⊢ (𝑈 ∈ Univ → (𝑅1‘𝐴) ⊆ 𝑈)
94	93	adantr 484	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → (𝑅1‘𝐴) ⊆ 𝑈)
95	92, 94	eqssd 3953	1 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → 𝑈 = (𝑅1‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∨ w3o 1096   = wceq 1559  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∃wrex 3085  Vcvv 3453   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4554  ∪ cuni 4864   class class class wbr 5099  Tr wtr 5206  dom cdm 5645   “ cima 5648  Ord word 6341  Oncon0 6342  Fun wfun 6511   Fn wfn 6512  ⟶wf 6513  ‘cfv 6517   ≼ cdom 8921   ≺ csdm 8922  TCctc 9686  𝑅₁cr1 9717  rankcrnk 9718  cfccf 9892  Inacc_wcwina 10637  Inacccina 10638  Univcgru 10745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593  ax-ac2 10417

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-iin 4951  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-tc 9687  df-r1 9719  df-rank 9720  df-card 9894  df-cf 9896  df-acn 9897  df-ac 10069  df-wina 10639  df-ina 10640  df-gru 10746

This theorem is referenced by:  grutsk  10777  bj-grur1  37512  grurankcld  44773