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Theorem rankonidlem 9819

Description: Lemma for rankonid 9820. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)

**Assertion**
Ref	Expression
rankonidlem	⊢ (𝐴 ∈ dom 𝑅₁ → (𝐴 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝐴) = 𝐴))

Proof of Theorem rankonidlem Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r1funlim 9757	. . . . 5 ⊢ (Fun 𝑅₁ ∧ Lim dom 𝑅₁)
2	1	simpri 486	. . . 4 ⊢ Lim dom 𝑅₁
3		limord 6421	. . . 4 ⊢ (Lim dom 𝑅₁ → Ord dom 𝑅₁)
4	2, 3	ax-mp 5	. . 3 ⊢ Ord dom 𝑅₁
5		ordelon 6385	. . 3 ⊢ ((Ord dom 𝑅₁ ∧ 𝐴 ∈ dom 𝑅₁) → 𝐴 ∈ On)
6	4, 5	mpan 688	. 2 ⊢ (𝐴 ∈ dom 𝑅₁ → 𝐴 ∈ On)
7		eleq1 2821	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅₁ ↔ 𝑦 ∈ dom 𝑅₁))
8		eleq1 2821	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∪ (𝑅₁ “ On) ↔ 𝑦 ∈ ∪ (𝑅₁ “ On)))
9		fveq2 6888	. . . . . 6 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
10		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
11	9, 10	eqeq12d 2748	. . . . 5 ⊢ (𝑥 = 𝑦 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝑦) = 𝑦))
12	8, 11	anbi12d 631	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)))
13	7, 12	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅₁ → (𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝑦 ∈ dom 𝑅₁ → (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦))))
14		eleq1 2821	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅₁ ↔ 𝐴 ∈ dom 𝑅₁))
15		eleq1 2821	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ∪ (𝑅₁ “ On) ↔ 𝐴 ∈ ∪ (𝑅₁ “ On)))
16		fveq2 6888	. . . . . 6 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
17		id 22	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
18	16, 17	eqeq12d 2748	. . . . 5 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝐴) = 𝐴))
19	15, 18	anbi12d 631	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝐴 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝐴) = 𝐴)))
20	14, 19	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ dom 𝑅₁ → (𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝐴 ∈ dom 𝑅₁ → (𝐴 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝐴) = 𝐴))))
21		ordtr1 6404	. . . . . . . . . 10 ⊢ (Ord dom 𝑅₁ → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅₁) → 𝑦 ∈ dom 𝑅₁))
22	4, 21	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅₁) → 𝑦 ∈ dom 𝑅₁)
23	22	ancoms 459	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝑅₁)
24		pm5.5 361	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑅₁ → ((𝑦 ∈ dom 𝑅₁ → (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)))
25	23, 24	syl 17	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ dom 𝑅₁ → (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)))
26	25	ralbidva 3175	. . . . . 6 ⊢ (𝑥 ∈ dom 𝑅₁ → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅₁ → (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)))
27		simplr 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ 𝑥)
28		ordelon 6385	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Ord dom 𝑅₁ ∧ 𝑥 ∈ dom 𝑅₁) → 𝑥 ∈ On)
29	4, 28	mpan 688	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ dom 𝑅₁ → 𝑥 ∈ On)
30	29	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
31		eloni 6371	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → Ord 𝑥)
32	30, 31	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → Ord 𝑥)
33		ordelsuc 7804	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑥 ∧ Ord 𝑥) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥))
34	27, 32, 33	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥))
35	27, 34	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ⊆ 𝑥)
36	23	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ dom 𝑅₁)
37		limsuc 7834	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim dom 𝑅₁ → (𝑦 ∈ dom 𝑅₁ ↔ suc 𝑦 ∈ dom 𝑅₁))
38	2, 37	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ dom 𝑅₁ ↔ suc 𝑦 ∈ dom 𝑅₁)
39	36, 38	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ∈ dom 𝑅₁)
40		simpll 765	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ dom 𝑅₁)
41		r1ord3g 9770	. . . . . . . . . . . . . . . . . 18 ⊢ ((suc 𝑦 ∈ dom 𝑅₁ ∧ 𝑥 ∈ dom 𝑅₁) → (suc 𝑦 ⊆ 𝑥 → (𝑅₁‘suc 𝑦) ⊆ (𝑅₁‘𝑥)))
42	39, 40, 41	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (suc 𝑦 ⊆ 𝑥 → (𝑅₁‘suc 𝑦) ⊆ (𝑅₁‘𝑥)))
43	35, 42	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅₁‘suc 𝑦) ⊆ (𝑅₁‘𝑥))
44		rankidb 9791	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ∪ (𝑅₁ “ On) → 𝑦 ∈ (𝑅₁‘suc (rank‘𝑦)))
45	44	ad2antrl 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅₁‘suc (rank‘𝑦)))
46		suceq 6427	. . . . . . . . . . . . . . . . . . 19 ⊢ ((rank‘𝑦) = 𝑦 → suc (rank‘𝑦) = suc 𝑦)
47	46	ad2antll 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → suc (rank‘𝑦) = suc 𝑦)
48	47	fveq2d 6892	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅₁‘suc (rank‘𝑦)) = (𝑅₁‘suc 𝑦))
49	45, 48	eleqtrd 2835	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅₁‘suc 𝑦))
50	43, 49	sseldd 3982	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅₁‘𝑥))
51	50	ex 413	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → 𝑦 ∈ (𝑅₁‘𝑥)))
52	51	ralimdva 3167	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑅₁ → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅₁‘𝑥)))
53	52	imp 407	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅₁‘𝑥))
54		dfss3 3969	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ (𝑅₁‘𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅₁‘𝑥))
55	53, 54	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ⊆ (𝑅₁‘𝑥))
56		vex 3478	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
57	56	elpw 4605	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 (𝑅₁‘𝑥) ↔ 𝑥 ⊆ (𝑅₁‘𝑥))
58	55, 57	sylibr 233	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ 𝒫 (𝑅₁‘𝑥))
59		r1sucg 9760	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝑅₁ → (𝑅₁‘suc 𝑥) = 𝒫 (𝑅₁‘𝑥))
60	59	adantr 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅₁‘suc 𝑥) = 𝒫 (𝑅₁‘𝑥))
61	58, 60	eleqtrrd 2836	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ (𝑅₁‘suc 𝑥))
62		r1elwf 9787	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑅₁‘suc 𝑥) → 𝑥 ∈ ∪ (𝑅₁ “ On))
63	61, 62	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ ∪ (𝑅₁ “ On))
64		rankval3b 9817	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅₁ “ On) → (rank‘𝑥) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧})
65	63, 64	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧})
66		eleq1 2821	. . . . . . . . . . . . . . . 16 ⊢ ((rank‘𝑦) = 𝑦 → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
67	66	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
68	67	ralimi 3083	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
69		ralbi 3103	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧))
70	68, 69	syl 17	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧))
71		dfss3 3969	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧)
72	70, 71	bitr4di 288	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ 𝑥 ⊆ 𝑧))
73	72	rabbidv 3440	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
74	73	inteqd 4954	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
75	74	adantl 482	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
76	29	adantr 481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
77		intmin 4971	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧} = 𝑥)
78	76, 77	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧} = 𝑥)
79	65, 75, 78	3eqtrd 2776	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = 𝑥)
80	63, 79	jca 512	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥))
81	80	ex 413	. . . . . 6 ⊢ (𝑥 ∈ dom 𝑅₁ → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → (𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥)))
82	26, 81	sylbid 239	. . . . 5 ⊢ (𝑥 ∈ dom 𝑅₁ → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅₁ → (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥)))
83	82	com12 32	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅₁ → (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ dom 𝑅₁ → (𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥)))
84	83	a1i 11	. . 3 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅₁ → (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ dom 𝑅₁ → (𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥))))
85	13, 20, 84	tfis3 7843	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom 𝑅₁ → (𝐴 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝐴) = 𝐴)))
86	6, 85	mpcom 38	1 ⊢ (𝐴 ∈ dom 𝑅₁ → (𝐴 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝐴) = 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {crab 3432 ⊆ wss 3947 𝒫 cpw 4601 ∪ cuni 4907 ∩ cint 4949 dom cdm 5675 “ cima 5678 Ord word 6360 Oncon0 6361 Lim wlim 6362 suc csuc 6363 Fun wfun 6534 ‘cfv 6540 𝑅₁cr1 9753 rankcrnk 9754

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-r1 9755 df-rank 9756

This theorem is referenced by: rankonid 9820 onwf 9821 onssr1 9822

W3C validator