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Theorem rankonidlem 9825

Description: Lemma for rankonid 9826. (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 9763	. . . . 5 ⊢ (Fun 𝑅₁ ∧ Lim dom 𝑅₁)
2	1	simpri 484	. . . 4 ⊢ Lim dom 𝑅₁
3		limord 6423	. . . 4 ⊢ (Lim dom 𝑅₁ → Ord dom 𝑅₁)
4	2, 3	ax-mp 5	. . 3 ⊢ Ord dom 𝑅₁
5		ordelon 6387	. . 3 ⊢ ((Ord dom 𝑅₁ ∧ 𝐴 ∈ dom 𝑅₁) → 𝐴 ∈ On)
6	4, 5	mpan 686	. 2 ⊢ (𝐴 ∈ dom 𝑅₁ → 𝐴 ∈ On)
7		eleq1 2819	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅₁ ↔ 𝑦 ∈ dom 𝑅₁))
8		eleq1 2819	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∪ (𝑅₁ “ On) ↔ 𝑦 ∈ ∪ (𝑅₁ “ On)))
9		fveq2 6890	. . . . . 6 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
10		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
11	9, 10	eqeq12d 2746	. . . . 5 ⊢ (𝑥 = 𝑦 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝑦) = 𝑦))
12	8, 11	anbi12d 629	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)))
13	7, 12	imbi12d 343	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅₁ → (𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝑦 ∈ dom 𝑅₁ → (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦))))
14		eleq1 2819	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅₁ ↔ 𝐴 ∈ dom 𝑅₁))
15		eleq1 2819	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ∪ (𝑅₁ “ On) ↔ 𝐴 ∈ ∪ (𝑅₁ “ On)))
16		fveq2 6890	. . . . . 6 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
17		id 22	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
18	16, 17	eqeq12d 2746	. . . . 5 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝐴) = 𝐴))
19	15, 18	anbi12d 629	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝐴 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝐴) = 𝐴)))
20	14, 19	imbi12d 343	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ dom 𝑅₁ → (𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝐴 ∈ dom 𝑅₁ → (𝐴 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝐴) = 𝐴))))
21		ordtr1 6406	. . . . . . . . . 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 457	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝑅₁)
24		pm5.5 360	. . . . . . . 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 3173	. . . . . 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 765	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ 𝑥)
28		ordelon 6387	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Ord dom 𝑅₁ ∧ 𝑥 ∈ dom 𝑅₁) → 𝑥 ∈ On)
29	4, 28	mpan 686	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ dom 𝑅₁ → 𝑥 ∈ On)
30	29	ad2antrr 722	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
31		eloni 6373	. . . . . . . . . . . . . . . . . . . 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 7810	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑥 ∧ Ord 𝑥) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥))
34	27, 32, 33	syl2anc 582	. . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ dom 𝑅₁)
37		limsuc 7840	. . . . . . . . . . . . . . . . . . . 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 763	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ dom 𝑅₁)
41		r1ord3g 9776	. . . . . . . . . . . . . . . . . 18 ⊢ ((suc 𝑦 ∈ dom 𝑅₁ ∧ 𝑥 ∈ dom 𝑅₁) → (suc 𝑦 ⊆ 𝑥 → (𝑅₁‘suc 𝑦) ⊆ (𝑅₁‘𝑥)))
42	39, 40, 41	syl2anc 582	. . . . . . . . . . . . . . . . 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 9797	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ∪ (𝑅₁ “ On) → 𝑦 ∈ (𝑅₁‘suc (rank‘𝑦)))
45	44	ad2antrl 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅₁‘suc (rank‘𝑦)))
46		suceq 6429	. . . . . . . . . . . . . . . . . . 19 ⊢ ((rank‘𝑦) = 𝑦 → suc (rank‘𝑦) = suc 𝑦)
47	46	ad2antll 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → suc (rank‘𝑦) = suc 𝑦)
48	47	fveq2d 6894	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅₁‘suc (rank‘𝑦)) = (𝑅₁‘suc 𝑦))
49	45, 48	eleqtrd 2833	. . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → 𝑦 ∈ (𝑅₁‘𝑥)))
52	51	ralimdva 3165	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑅₁ → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅₁‘𝑥)))
53	52	imp 405	. . . . . . . . . . . 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 3476	. . . . . . . . . . . 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 9766	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝑅₁ → (𝑅₁‘suc 𝑥) = 𝒫 (𝑅₁‘𝑥))
60	59	adantr 479	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅₁‘suc 𝑥) = 𝒫 (𝑅₁‘𝑥))
61	58, 60	eleqtrrd 2834	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ (𝑅₁‘suc 𝑥))
62		r1elwf 9793	. . . . . . . . 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 9823	. . . . . . . . . 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 2819	. . . . . . . . . . . . . . . 16 ⊢ ((rank‘𝑦) = 𝑦 → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
67	66	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
68	67	ralimi 3081	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
69		ralbi 3101	. . . . . . . . . . . . . 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 3438	. . . . . . . . . . 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 480	. . . . . . . . 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 479	. . . . . . . . . 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 2774	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = 𝑥)
80	63, 79	jca 510	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝑅₁ ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪ (𝑅₁ “ On) ∧ (rank‘𝑥) = 𝑥))
81	80	ex 411	. . . . . 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 7849	. 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 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 {crab 3430 ⊆ wss 3947 𝒫 cpw 4601 ∪ cuni 4907 ∩ cint 4949 dom cdm 5675 “ cima 5678 Ord word 6362 Oncon0 6363 Lim wlim 6364 suc csuc 6365 Fun wfun 6536 ‘cfv 6542 𝑅₁cr1 9759 rankcrnk 9760

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-r1 9761 df-rank 9762

This theorem is referenced by: rankonid 9826 onwf 9827 onssr1 9828

W3C validator