Theorem rankonidlem 9838

Description: Lemma for rankonid 9839. (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		r1dmlim 9772	. . . 4 ⊢ Lim dom 𝑅1
2		limord 6424	. . . 4 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1)
3	1, 2	ax-mp 5	. . 3 ⊢ Ord dom 𝑅1
4		ordelon 6386	. . 3 ⊢ ((Ord dom 𝑅1 ∧ 𝐴 ∈ dom 𝑅1) → 𝐴 ∈ On)
5	3, 4	mpan 703	. 2 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On)
6		eleq1 2848	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ dom 𝑅1 ↔ 𝑦 ∈ dom 𝑅1))
7		eleq1 2848	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑦 ∈ ∪ (𝑅1 “ On)))
8		fveq2 6885	. . . . . 6 ⊢ (𝑥 = 𝑦 → (rank‘𝑥) = (rank‘𝑦))
9		id 23	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
10	8, 9	eqeq12d 2776	. . . . 5 ⊢ (𝑥 = 𝑦 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝑦) = 𝑦))
11	7, 10	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
12	6, 11	imbi12d 347	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ dom 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦))))
13		eleq1 2848	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝑅1 ↔ 𝐴 ∈ dom 𝑅1))
14		eleq1 2848	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ∈ ∪ (𝑅1 “ On)))
15		fveq2 6885	. . . . . 6 ⊢ (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
16		id 23	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
17	15, 16	eqeq12d 2776	. . . . 5 ⊢ (𝑥 = 𝐴 → ((rank‘𝑥) = 𝑥 ↔ (rank‘𝐴) = 𝐴))
18	14, 17	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥) ↔ (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)))
19	13, 18	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ dom 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) ↔ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))))
20		ordtr1 6407	. . . . . . . . . 10 ⊢ (Ord dom 𝑅1 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → 𝑦 ∈ dom 𝑅1))
21	3, 20	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝑅1) → 𝑦 ∈ dom 𝑅1)
22	21	ancoms 464	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝑅1)
23		pm5.5 364	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑅1 → ((𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
24	22, 23	syl 18	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
25	24	ralbidva 3183	. . . . . 6 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)))
26		simplr 781	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ 𝑥)
27		ordelon 6386	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Ord dom 𝑅1 ∧ 𝑥 ∈ dom 𝑅1) → 𝑥 ∈ On)
28	3, 27	mpan 703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ dom 𝑅1 → 𝑥 ∈ On)
29	28	ad2antrr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
30		eloni 6372	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ On → Ord 𝑥)
31	29, 30	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → Ord 𝑥)
32		ordelsuc 7831	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑥 ∧ Ord 𝑥) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥))
33	26, 31, 32	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑦 ∈ 𝑥 ↔ suc 𝑦 ⊆ 𝑥))
34	26, 33	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ⊆ 𝑥)
35	22	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ dom 𝑅1)
36		limsuc 7860	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim dom 𝑅1 → (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1))
37	1, 36	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ dom 𝑅1 ↔ suc 𝑦 ∈ dom 𝑅1)
38	35, 37	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc 𝑦 ∈ dom 𝑅1)
39		simpll 779	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ dom 𝑅1)
40		r1ord3g 9786	. . . . . . . . . . . . . . . . . 18 ⊢ ((suc 𝑦 ∈ dom 𝑅1 ∧ 𝑥 ∈ dom 𝑅1) → (suc 𝑦 ⊆ 𝑥 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥)))
41	38, 39, 40	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (suc 𝑦 ⊆ 𝑥 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥)))
42	34, 41	mpd 16	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝑥))
43		rankidb 9808	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ∪ (𝑅1 “ On) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
44	43	ad2antrl 741	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc (rank‘𝑦)))
45		suceq 6431	. . . . . . . . . . . . . . . . . . 19 ⊢ ((rank‘𝑦) = 𝑦 → suc (rank‘𝑦) = suc 𝑦)
46	45	ad2antll 742	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → suc (rank‘𝑦) = suc 𝑦)
47	46	fveq2d 6889	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc (rank‘𝑦)) = (𝑅1‘suc 𝑦))
48	44, 47	eleqtrd 2862	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘suc 𝑦))
49	42, 48	sseldd 3909	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) ∧ (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑦 ∈ (𝑅1‘𝑥))
50	49	ex 418	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ 𝑦 ∈ 𝑥) → ((𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → 𝑦 ∈ (𝑅1‘𝑥)))
51	50	ralimdva 3174	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥)))
52	51	imp 412	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥))
53		dfss3 3897	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ (𝑅1‘𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝑅1‘𝑥))
54	52, 53	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ⊆ (𝑅1‘𝑥))
55		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
56	55	elpw 4560	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 (𝑅1‘𝑥) ↔ 𝑥 ⊆ (𝑅1‘𝑥))
57	54, 56	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ 𝒫 (𝑅1‘𝑥))
58		r1sucg 9776	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
59	58	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑅1‘suc 𝑥) = 𝒫 (𝑅1‘𝑥))
60	57, 59	eleqtrrd 2863	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ (𝑅1‘suc 𝑥))
61		r1elwf 9804	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑅1‘suc 𝑥) → 𝑥 ∈ ∪ (𝑅1 “ On))
62	60, 61	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ ∪ (𝑅1 “ On))
63		rankval3b 9836	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (rank‘𝑥) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧})
64	62, 63	syl 18	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧})
65		eleq1 2848	. . . . . . . . . . . . . . . 16 ⊢ ((rank‘𝑦) = 𝑦 → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
66	65	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
67	66	ralimi 3099	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
68		ralbi 3117	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 ((rank‘𝑦) ∈ 𝑧 ↔ 𝑦 ∈ 𝑧) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧))
69	67, 68	syl 18	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧))
70		dfss3 3897	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑧 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑧)
71	69, 70	bitr4di 292	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧 ↔ 𝑥 ⊆ 𝑧))
72	71	rabbidv 3419	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
73	72	inteqd 4911	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
74	73	adantl 487	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∩ {𝑧 ∈ On ∣ ∀𝑦 ∈ 𝑥 (rank‘𝑦) ∈ 𝑧} = ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧})
75	28	adantr 486	. . . . . . . . . 10 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → 𝑥 ∈ On)
76		intmin 4927	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧} = 𝑥)
77	75, 76	syl 18	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → ∩ {𝑧 ∈ On ∣ 𝑥 ⊆ 𝑧} = 𝑥)
78	64, 74, 77	3eqtrd 2799	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (rank‘𝑥) = 𝑥)
79	62, 78	jca 521	. . . . . . 7 ⊢ ((𝑥 ∈ dom 𝑅1 ∧ ∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥))
80	79	ex 418	. . . . . 6 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦) → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
81	25, 80	sylbid 243	. . . . 5 ⊢ (𝑥 ∈ dom 𝑅1 → (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
82	81	com12 33	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ dom 𝑅1 → (𝑦 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑦) = 𝑦)) → (𝑥 ∈ dom 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)))
83	82	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))))
84	12, 19, 83	tfis3 7869	. 2 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)))
85	5, 84	mpcom 39	1 ⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3895  𝒫 cpw 4556  ∪ cuni 4866  ∩ cint 4906  dom cdm 5650   “ cima 5653  Ord word 6361  Oncon0 6362  Lim wlim 6363  suc csuc 6364  ‘cfv 6538  𝑅₁cr1 9766  rankcrnk 9767

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768  df-rank 9769

This theorem is used by:  rankonid  9839  onwf  9840  onssr1  9843