Theorem grur1 10576

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 3983	. . . . 5 ⊢ (¬ 𝑈 ⊆ (𝑅1‘𝐴) ↔ ∃𝑥(𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴)))
2		fveqeq2 6783	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((rank‘𝑦) = 𝐴 ↔ (rank‘𝑥) = 𝐴))
3	2	rspcev 3561	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑈 ∧ (rank‘𝑥) = 𝐴) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴)
4	3	ex 413	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → ((rank‘𝑥) = 𝐴 → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
5	4	ad2antrl 725	. . . . . . . 8 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ((rank‘𝑥) = 𝐴 → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
6		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → 𝑈 ∈ ∪ (𝑅1 “ On))
7		simprl 768	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → 𝑥 ∈ 𝑈)
8		r1elssi 9563	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ ∪ (𝑅1 “ On) → 𝑈 ⊆ ∪ (𝑅1 “ On))
9	8	sseld 3920	. . . . . . . . . . . 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 9642	. . . . . . . . . . 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 10550	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ 𝑈)
15		grutr 10549	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ Univ → Tr 𝑈)
16	15	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → Tr 𝑈)
17		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
18		tcmin 9499	. . . . . . . . . . . . 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 584	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → (TC‘𝑥) ⊆ 𝑈)
21		rankf 9552	. . . . . . . . . . . . 13 ⊢ rank:∪ (𝑅1 “ On)⟶On
22		ffun 6603	. . . . . . . . . . . . 13 ⊢ (rank:∪ (𝑅1 “ On)⟶On → Fun rank)
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun rank
24		fvelima 6835	. . . . . . . . . . . 12 ⊢ ((Fun rank ∧ 𝐴 ∈ (rank “ (TC‘𝑥))) → ∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴)
25	23, 24	mpan 687	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴)
26		ssrexv 3988	. . . . . . . . . . 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 744	. . . . . . . . 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 770	. . . . . . . . . 10 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ¬ 𝑥 ∈ (𝑅1‘𝐴))
31		ne0i 4268	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑈 → 𝑈 ≠ ∅)
32		gruina.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = (𝑈 ∩ On)
33	32	gruina 10574	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝐴 ∈ Inacc)
34	31, 33	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ Inacc)
35		inawina 10446	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
36		winaon 10444	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ On)
37	34, 35, 36	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ On)
38		r1fnon 9525	. . . . . . . . . . . . . 14 ⊢ 𝑅1 Fn On
39		fndm 6536	. . . . . . . . . . . . . 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 744	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → 𝐴 ∈ dom 𝑅1)
43		rankr1ag 9560	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
44	10, 42, 43	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
45	30, 44	mtbid 324	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ¬ (rank‘𝑥) ∈ 𝐴)
46		rankon 9553	. . . . . . . . . . . . 13 ⊢ (rank‘𝑥) ∈ On
47		eloni 6276	. . . . . . . . . . . . . 14 ⊢ ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
48		eloni 6276	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → Ord 𝐴)
49		ordtri3or 6298	. . . . . . . . . . . . . 14 ⊢ ((Ord (rank‘𝑥) ∧ Ord 𝐴) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
50	47, 48, 49	syl2an 596	. . . . . . . . . . . . 13 ⊢ (((rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
51	46, 37, 50	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
52		3orass 1089	. . . . . . . . . . . 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 861	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → (¬ (rank‘𝑥) ∈ 𝐴 → ((rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥))))
55	54	ad2ant2r 744	. . . . . . . . 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 857	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴))) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴)
58	57	ex 413	. . . . . 6 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → ((𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅1‘𝐴)) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
59	58	exlimdv 1936	. . . . 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 764	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ∈ Univ)
62		ne0i 4268	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑈 → 𝑈 ≠ ∅)
63	62, 33	sylan2 593	. . . . . . . . 9 ⊢ ((𝑈 ∈ Univ ∧ 𝑦 ∈ 𝑈) → 𝐴 ∈ Inacc)
64	63	ad2ant2r 744	. . . . . . . 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 768	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑦 ∈ 𝑈)
67		fveq2 6774	. . . . . . . . . 10 ⊢ ((rank‘𝑦) = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘𝐴))
68	67	ad2antll 726	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘(rank‘𝑦)) = (cf‘𝐴))
69		elina 10443	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
70	69	simp2bi 1145	. . . . . . . . . 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 10533	. . . . . . . . 9 ⊢ ¬ 𝑦 ≺ (cf‘(rank‘𝑦))
74		fvex 6787	. . . . . . . . . 10 ⊢ (cf‘(rank‘𝑦)) ∈ V
75		vex 3436	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
76		domtri 10312	. . . . . . . . . 10 ⊢ (((cf‘(rank‘𝑦)) ∈ V ∧ 𝑦 ∈ V) → ((cf‘(rank‘𝑦)) ≼ 𝑦 ↔ ¬ 𝑦 ≺ (cf‘(rank‘𝑦))))
77	74, 75, 76	mp2an 689	. . . . . . . . 9 ⊢ ((cf‘(rank‘𝑦)) ≼ 𝑦 ↔ ¬ 𝑦 ≺ (cf‘(rank‘𝑦)))
78	73, 77	mpbir 230	. . . . . . . 8 ⊢ (cf‘(rank‘𝑦)) ≼ 𝑦
79	72, 78	eqbrtrrdi 5114	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ≼ 𝑦)
80		grudomon 10573	. . . . . . 7 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝑦 ∈ 𝑈 ∧ 𝐴 ≼ 𝑦)) → 𝐴 ∈ 𝑈)
81	61, 65, 66, 79, 80	syl112anc 1373	. . . . . 6 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ∈ 𝑈)
82		elin 3903	. . . . . . . . 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 6284	. . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
86	48, 85	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ 𝐴 ∈ 𝐴)
87	86	adantl 482	. . . . . . 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 584	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ⊆ (𝑅1‘𝐴))
90	89	rexlimdvaa 3214	. . . 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 10575	. . 3 ⊢ (𝑈 ∈ Univ → (𝑅1‘𝐴) ⊆ 𝑈)
94	93	adantr 481	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → (𝑅1‘𝐴) ⊆ 𝑈)
95	92, 94	eqssd 3938	1 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅1 “ On)) → 𝑈 = (𝑅1‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∨ w3o 1085   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  ∪ cuni 4839   class class class wbr 5074  Tr wtr 5191  dom cdm 5589   “ cima 5592  Ord word 6265  Oncon0 6266  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433   ≼ cdom 8731   ≺ csdm 8732  TCctc 9494  𝑅₁cr1 9520  rankcrnk 9521  cfccf 9695  Inacc_wcwina 10438  Inacccina 10439  Univcgru 10546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-ac2 10219

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-tc 9495  df-r1 9522  df-rank 9523  df-card 9697  df-cf 9699  df-acn 9700  df-ac 9872  df-wina 10440  df-ina 10441  df-gru 10547

This theorem is referenced by:  grutsk  10578  bj-grur1  35234  grurankcld  41851