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Theorem grur1 10812

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 4046	. . . . 5 ⊢ (¬ 𝑈 ⊆ (𝑅₁‘𝐴) ↔ ∃𝑥(𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅₁‘𝐴)))
2		fveqeq2 6898	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((rank‘𝑦) = 𝐴 ↔ (rank‘𝑥) = 𝐴))
3	2	rspcev 3613	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑈 ∧ (rank‘𝑥) = 𝐴) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴)
4	3	ex 414	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → ((rank‘𝑥) = 𝐴 → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
5	4	ad2antrl 727	. . . . . . . 8 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅₁‘𝐴))) → ((rank‘𝑥) = 𝐴 → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
6		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅₁‘𝐴))) → 𝑈 ∈ ∪ (𝑅₁ “ On))
7		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅₁‘𝐴))) → 𝑥 ∈ 𝑈)
8		r1elssi 9797	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ ∪ (𝑅₁ “ On) → 𝑈 ⊆ ∪ (𝑅₁ “ On))
9	8	sseld 3981	. . . . . . . . . . . 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 9876	. . . . . . . . . . 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 2820	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅₁‘𝐴))) → (𝐴 ∈ (rank‘𝑥) ↔ 𝐴 ∈ (rank “ (TC‘𝑥))))
14		gruelss 10786	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝑥 ⊆ 𝑈)
15		grutr 10785	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ Univ → Tr 𝑈)
16	15	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → Tr 𝑈)
17		vex 3479	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
18		tcmin 9733	. . . . . . . . . . . . 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 585	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → (TC‘𝑥) ⊆ 𝑈)
21		rankf 9786	. . . . . . . . . . . . 13 ⊢ rank:∪ (𝑅₁ “ On)⟶On
22		ffun 6718	. . . . . . . . . . . . 13 ⊢ (rank:∪ (𝑅₁ “ On)⟶On → Fun rank)
23	21, 22	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun rank
24		fvelima 6955	. . . . . . . . . . . 12 ⊢ ((Fun rank ∧ 𝐴 ∈ (rank “ (TC‘𝑥))) → ∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴)
25	23, 24	mpan 689	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (rank “ (TC‘𝑥)) → ∃𝑦 ∈ (TC‘𝑥)(rank‘𝑦) = 𝐴)
26		ssrexv 4051	. . . . . . . . . . 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 746	. . . . . . . . 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 772	. . . . . . . . . 10 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅₁‘𝐴))) → ¬ 𝑥 ∈ (𝑅₁‘𝐴))
31		ne0i 4334	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑈 → 𝑈 ≠ ∅)
32		gruina.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐴 = (𝑈 ∩ On)
33	32	gruina 10810	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ≠ ∅) → 𝐴 ∈ Inacc)
34	31, 33	sylan2 594	. . . . . . . . . . . . . 14 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ Inacc)
35		inawina 10682	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inacc_w)
36		winaon 10680	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Inacc_w → 𝐴 ∈ On)
37	34, 35, 36	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ On)
38		r1fnon 9759	. . . . . . . . . . . . . 14 ⊢ 𝑅₁ Fn On
39		fndm 6650	. . . . . . . . . . . . . 14 ⊢ (𝑅₁ Fn On → dom 𝑅₁ = On)
40	38, 39	ax-mp 5	. . . . . . . . . . . . 13 ⊢ dom 𝑅₁ = On
41	37, 40	eleqtrrdi 2845	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → 𝐴 ∈ dom 𝑅₁)
42	41	ad2ant2r 746	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅₁‘𝐴))) → 𝐴 ∈ dom 𝑅₁)
43		rankr1ag 9794	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ∈ dom 𝑅₁) → (𝑥 ∈ (𝑅₁‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
44	10, 42, 43	syl2anc 585	. . . . . . . . . 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 9787	. . . . . . . . . . . . 13 ⊢ (rank‘𝑥) ∈ On
47		eloni 6372	. . . . . . . . . . . . . 14 ⊢ ((rank‘𝑥) ∈ On → Ord (rank‘𝑥))
48		eloni 6372	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → Ord 𝐴)
49		ordtri3or 6394	. . . . . . . . . . . . . 14 ⊢ ((Ord (rank‘𝑥) ∧ Ord 𝐴) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
50	47, 48, 49	syl2an 597	. . . . . . . . . . . . 13 ⊢ (((rank‘𝑥) ∈ On ∧ 𝐴 ∈ On) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
51	46, 37, 50	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥)))
52		3orass 1091	. . . . . . . . . . . 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 863	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Univ ∧ 𝑥 ∈ 𝑈) → (¬ (rank‘𝑥) ∈ 𝐴 → ((rank‘𝑥) = 𝐴 ∨ 𝐴 ∈ (rank‘𝑥))))
55	54	ad2ant2r 746	. . . . . . . . 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 859	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ (𝑅₁‘𝐴))) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴)
58	57	ex 414	. . . . . 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	biimtrid 241	. . . 4 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) → (¬ 𝑈 ⊆ (𝑅₁‘𝐴) → ∃𝑦 ∈ 𝑈 (rank‘𝑦) = 𝐴))
61		simpll 766	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ∈ Univ)
62		ne0i 4334	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑈 → 𝑈 ≠ ∅)
63	62, 33	sylan2 594	. . . . . . . . 9 ⊢ ((𝑈 ∈ Univ ∧ 𝑦 ∈ 𝑈) → 𝐴 ∈ Inacc)
64	63	ad2ant2r 746	. . . . . . . 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 770	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑦 ∈ 𝑈)
67		fveq2 6889	. . . . . . . . . 10 ⊢ ((rank‘𝑦) = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘𝐴))
68	67	ad2antll 728	. . . . . . . . 9 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘(rank‘𝑦)) = (cf‘𝐴))
69		elina 10679	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
70	69	simp2bi 1147	. . . . . . . . . 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 2773	. . . . . . . 8 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → (cf‘(rank‘𝑦)) = 𝐴)
73		rankcf 10769	. . . . . . . . 9 ⊢ ¬ 𝑦 ≺ (cf‘(rank‘𝑦))
74		fvex 6902	. . . . . . . . . 10 ⊢ (cf‘(rank‘𝑦)) ∈ V
75		vex 3479	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
76		domtri 10548	. . . . . . . . . 10 ⊢ (((cf‘(rank‘𝑦)) ∈ V ∧ 𝑦 ∈ V) → ((cf‘(rank‘𝑦)) ≼ 𝑦 ↔ ¬ 𝑦 ≺ (cf‘(rank‘𝑦))))
77	74, 75, 76	mp2an 691	. . . . . . . . 9 ⊢ ((cf‘(rank‘𝑦)) ≼ 𝑦 ↔ ¬ 𝑦 ≺ (cf‘(rank‘𝑦)))
78	73, 77	mpbir 230	. . . . . . . 8 ⊢ (cf‘(rank‘𝑦)) ≼ 𝑦
79	72, 78	eqbrtrrdi 5188	. . . . . . 7 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ≼ 𝑦)
80		grudomon 10809	. . . . . . 7 ⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ On ∧ (𝑦 ∈ 𝑈 ∧ 𝐴 ≼ 𝑦)) → 𝐴 ∈ 𝑈)
81	61, 65, 66, 79, 80	syl112anc 1375	. . . . . 6 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝐴 ∈ 𝑈)
82		elin 3964	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝑈 ∩ On) ↔ (𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On))
83	82	biimpri 227	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On) → 𝐴 ∈ (𝑈 ∩ On))
84	83, 32	eleqtrrdi 2845	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐴 ∈ On) → 𝐴 ∈ 𝐴)
85		ordirr 6380	. . . . . . . . 9 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
86	48, 85	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ 𝐴 ∈ 𝐴)
87	86	adantl 483	. . . . . . 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 585	. . . . 5 ⊢ (((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) ∧ (𝑦 ∈ 𝑈 ∧ (rank‘𝑦) = 𝐴)) → 𝑈 ⊆ (𝑅₁‘𝐴))
90	89	rexlimdvaa 3157	. . . 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 10811	. . 3 ⊢ (𝑈 ∈ Univ → (𝑅₁‘𝐴) ⊆ 𝑈)
94	93	adantr 482	. 2 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) → (𝑅₁‘𝐴) ⊆ 𝑈)
95	92, 94	eqssd 3999	1 ⊢ ((𝑈 ∈ Univ ∧ 𝑈 ∈ ∪ (𝑅₁ “ On)) → 𝑈 = (𝑅₁‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∨ w3o 1087 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 ∪ cuni 4908 class class class wbr 5148 Tr wtr 5265 dom cdm 5676 “ cima 5679 Ord word 6361 Oncon0 6362 Fun wfun 6535 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 ≼ cdom 8934 ≺ csdm 8935 TCctc 9728 𝑅₁cr1 9754 rankcrnk 9755 cfccf 9929 Inacc_wcwina 10674 Inacccina 10675 Univcgru 10782

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-ac2 10455

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-tc 9729 df-r1 9756 df-rank 9757 df-card 9931 df-cf 9933 df-acn 9934 df-ac 10108 df-wina 10676 df-ina 10677 df-gru 10783

This theorem is referenced by: grutsk 10814 bj-grur1 35933 grurankcld 42978

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