Theorem grur1 10507

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 3979	. . . . 5 ⊢ (¬ 𝑈 ⊆ (𝑅1‘𝐴) ↔ ∃𝑥(𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴)))
2		fveqeq2 6765	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((rank‘𝑦) = 𝐴 ↔ (rank‘𝑥) = 𝐴))
3	2	rspcev 3552	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑈 ∧ (rank‘𝑥) = 𝐴) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴)
4	3	ex 412	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → ((rank‘𝑥) = 𝐴 → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
5	4	ad2antrl 724	. . . . . . . 8 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ((rank‘𝑥) = 𝐴 → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
6		simplr 765	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → 𝑈 ∈ ∪ (𝑅1 “ On))
7		simprl 767	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → 𝑥 ∈ 𝑈)
8		r1elssi 9494	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ ∪ (𝑅1 “ On) → 𝑈 ⊆ ∪ (𝑅1 “ On))
9	8	sseld 3916	. . . . . . . . . . . 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 9573	. . . . . . . . . . 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 2824	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → (𝐴 ∈ (rank‘𝑥) ↔ 𝐴 ∈ (rank “ (TC‘𝑥))))
14		gruelss 10481	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ 𝑈)
15		grutr 10480	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ Univ → Tr 𝑈)
16	15	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → Tr 𝑈)
17		vex 3426	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
18		tcmin 9430	. . . . . . . . . . . . 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 583	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → (TC‘𝑥) ⊆ 𝑈)
21		rankf 9483	. . . . . . . . . . . . 13 ⊢ rank:∪ (𝑅1 “ On)⟶On
22		ffun 6587	. . . . . . . . . . . . 13 ⊢ (rank:∪ (𝑅1 “ On)⟶On → Fun rank)
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun rank
24		fvelima 6817	. . . . . . . . . . . 12 ⊢ ((Fun rank ∧ 𝐴 ∈ (rank “ (TC‘𝑥))) → ∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴)
25	23, 24	mpan 686	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴)
26		ssrexv 3984	. . . . . . . . . . 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 743	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
29	13, 28	sylbid 239	. . . . . . . 8 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → (𝐴 ∈ (rank‘𝑥) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
30		simprr 769	. . . . . . . . . 10 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ¬ 𝑥 ∈ (𝑅1‘𝐴))
31		ne0i 4265	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑈 → 𝑈 ≠ ∅)
32		gruina.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = (𝑈 ∩ On)
33	32	gruina 10505	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝐴 ∈ Inacc)
34	31, 33	sylan2 592	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ Inacc)
35		inawina 10377	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
36		winaon 10375	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ On)
37	34, 35, 36	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ On)
38		r1fnon 9456	. . . . . . . . . . . . . 14 ⊢ 𝑅1 Fn On
39		fndm 6520	. . . . . . . . . . . . . 14 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
40	38, 39	ax-mp 5	. . . . . . . . . . . . 13 ⊢ dom 𝑅1 = On
41	37, 40	eleqtrrdi 2850	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ dom 𝑅1)
42	41	ad2ant2r 743	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → 𝐴 ∈ dom 𝑅1)
43		rankr1ag 9491	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
44	10, 42, 43	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
45	30, 44	mtbid 323	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ¬ (rank‘𝑥) ∈ 𝐴)
46		rankon 9484	. . . . . . . . . . . . 13 ⊢ (rank‘𝑥) ∈ On
47		eloni 6261	. . . . . . . . . . . . . 14 ⊢ ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
48		eloni 6261	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → Ord 𝐴)
49		ordtri3or 6283	. . . . . . . . . . . . . 14 ⊢ ((Ord (rank‘𝑥) ∧ Ord 𝐴) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
50	47, 48, 49	syl2an 595	. . . . . . . . . . . . 13 ⊢ (((rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
51	46, 37, 50	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
52		3orass 1088	. . . . . . . . . . . 12 ⊢ (((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)) ↔ ((rank‘𝑥) ∈ 𝐴 ∨ ((rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥))))
53	51, 52	sylib 217	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → ((rank‘𝑥) ∈ 𝐴 ∨ ((rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥))))
54	53	ord 860	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → (¬ (rank‘𝑥) ∈ 𝐴 → ((rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥))))
55	54	ad2ant2r 743	. . . . . . . . 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 856	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴)
58	57	ex 412	. . . . . 6 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → ((𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴)) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
59	58	exlimdv 1937	. . . . 5 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → (∃𝑥(𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴)) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
60	1, 59	syl5bi 241	. . . 4 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → (¬ 𝑈 ⊆ (𝑅1‘𝐴) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
61		simpll 763	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ∈ Univ)
62		ne0i 4265	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑈 → 𝑈 ≠ ∅)
63	62, 33	sylan2 592	. . . . . . . . 9 ⊢ ((𝑈 ∈ Univ ∧ 𝑦 ∈ 𝑈) → 𝐴 ∈ Inacc)
64	63	ad2ant2r 743	. . . . . . . 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 767	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑦 ∈ 𝑈)
67		fveq2 6756	. . . . . . . . . 10 ⊢ ((rank‘𝑦) = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘𝐴))
68	67	ad2antll 725	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘(rank‘𝑦)) = (cf‘𝐴))
69		elina 10374	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
70	69	simp2bi 1144	. . . . . . . . . 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 2778	. . . . . . . 8 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘(rank‘𝑦)) = 𝐴)
73		rankcf 10464	. . . . . . . . 9 ⊢ ¬ 𝑦 ≺ (cf‘(rank‘𝑦))
74		fvex 6769	. . . . . . . . . 10 ⊢ (cf‘(rank‘𝑦)) ∈ V
75		vex 3426	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
76		domtri 10243	. . . . . . . . . 10 ⊢ (((cf‘(rank‘𝑦)) ∈ V ∧ 𝑦 ∈ V) → ((cf‘(rank‘𝑦)) ≼ 𝑦 ↔ ¬ 𝑦 ≺ (cf‘(rank‘𝑦))))
77	74, 75, 76	mp2an 688	. . . . . . . . 9 ⊢ ((cf‘(rank‘𝑦)) ≼ 𝑦 ↔ ¬ 𝑦 ≺ (cf‘(rank‘𝑦)))
78	73, 77	mpbir 230	. . . . . . . 8 ⊢ (cf‘(rank‘𝑦)) ≼ 𝑦
79	72, 78	eqbrtrrdi 5110	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ≼ 𝑦)
80		grudomon 10504	. . . . . . 7 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝑦 ∈ 𝑈 ∧ 𝐴 ≼ 𝑦)) → 𝐴 ∈ 𝑈)
81	61, 65, 66, 79, 80	syl112anc 1372	. . . . . 6 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ∈ 𝑈)
82		elin 3899	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑈 ∩ On) ↔ (𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On))
83	82	biimpri 227	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On) → 𝐴 ∈ (𝑈 ∩ On))
84	83, 32	eleqtrrdi 2850	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On) → 𝐴 ∈ 𝐴)
85		ordirr 6269	. . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
86	48, 85	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ 𝐴 ∈ 𝐴)
87	86	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On) → ¬ 𝐴 ∈ 𝐴)
88	84, 87	pm2.21dd 194	. . . . . 6 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On) → 𝑈 ⊆ (𝑅1‘𝐴))
89	81, 65, 88	syl2anc 583	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ⊆ (𝑅1‘𝐴))
90	89	rexlimdvaa 3213	. . . 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 10506	. . 3 ⊢ (𝑈 ∈ Univ → (𝑅1‘𝐴) ⊆ 𝑈)
94	93	adantr 480	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → (𝑅1‘𝐴) ⊆ 𝑈)
95	92, 94	eqssd 3934	1 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → 𝑈 = (𝑅1‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∨ w3o 1084   = wceq 1539  ∃wex 1783   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070  Tr wtr 5187  dom cdm 5580   “ cima 5583  Ord word 6250  Oncon0 6251  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418   ≼ cdom 8689   ≺ csdm 8690  TCctc 9425  𝑅₁cr1 9451  rankcrnk 9452  cfccf 9626  Inacc_wcwina 10369  Inacccina 10370  Univcgru 10477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-ac2 10150

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-tc 9426  df-r1 9453  df-rank 9454  df-card 9628  df-cf 9630  df-acn 9631  df-ac 9803  df-wina 10371  df-ina 10372  df-gru 10478

This theorem is referenced by:  grutsk  10509  bj-grur1  35161  grurankcld  41740