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Theorem rankxplim3 9791

Description: The rank of a Cartesian product is a limit ordinal iff its union is. (Contributed by NM, 19-Sep-2006.)

**Hypotheses**
Ref	Expression
rankxplim.1	⊢ 𝐴 ∈ V
rankxplim.2	⊢ 𝐵 ∈ V

**Assertion**
Ref	Expression
rankxplim3	⊢ (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim ∪ (rank‘(𝐴 × 𝐵)))

Proof of Theorem rankxplim3 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limuni2 6378	. 2 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim ∪ (rank‘(𝐴 × 𝐵)))
2		0ellim 6379	. . . 4 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ∅ ∈ ∪ (rank‘(𝐴 × 𝐵)))
3		n0i 4290	. . . 4 ⊢ (∅ ∈ ∪ (rank‘(𝐴 × 𝐵)) → ¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
4		unieq 4872	. . . . . 6 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ → ∪ (rank‘(𝐴 × 𝐵)) = ∪ ∅)
5		uni0 4889	. . . . . 6 ⊢ ∪ ∅ = ∅
6	4, 5	eqtrdi 2785	. . . . 5 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ → ∪ (rank‘(𝐴 × 𝐵)) = ∅)
7	6	con3i 154	. . . 4 ⊢ (¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅ → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
8	2, 3, 7	3syl 18	. . 3 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ¬ (rank‘(𝐴 × 𝐵)) = ∅)
9		rankon 9705	. . . . . . . . . 10 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On
10	9	onsuci 7779	. . . . . . . . 9 ⊢ suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
11	10	onsuci 7779	. . . . . . . 8 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
12	11	elexi 3461	. . . . . . 7 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ V
13	12	sucid 6399	. . . . . 6 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc suc suc (rank‘(𝐴 ∪ 𝐵))
14	11	onsuci 7779	. . . . . . . 8 ⊢ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
15		ontri1 6349	. . . . . . . 8 ⊢ ((suc suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On ∧ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On) → (suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ ¬ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc suc suc (rank‘(𝐴 ∪ 𝐵))))
16	14, 11, 15	mp2an 692	. . . . . . 7 ⊢ (suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ ¬ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc suc suc (rank‘(𝐴 ∪ 𝐵)))
17	16	con2bii 357	. . . . . 6 ⊢ (suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ↔ ¬ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)))
18	13, 17	mpbi 230	. . . . 5 ⊢ ¬ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))
19		rankxplim.1	. . . . . . 7 ⊢ 𝐴 ∈ V
20		rankxplim.2	. . . . . . 7 ⊢ 𝐵 ∈ V
21	19, 20	rankxpu 9786	. . . . . 6 ⊢ (rank‘(𝐴 × 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))
22		sstr 3940	. . . . . 6 ⊢ ((suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)) ∧ (rank‘(𝐴 × 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)))
23	21, 22	mpan2 691	. . . . 5 ⊢ (suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)))
24	18, 23	mto 197	. . . 4 ⊢ ¬ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))
25		reeanv 3206	. . . . 5 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ On ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦) ↔ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦))
26		simprl 770	. . . . . . . . . . . . 13 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥)
27		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥)
28		df-ne 2931	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅)
29	19, 20	xpex 7696	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 × 𝐵) ∈ V
30	29	rankeq0 9771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
31	30	notbii 320	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ (𝐴 × 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
32	28, 31	bitr2i 276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ ↔ (𝐴 × 𝐵) ≠ ∅)
33	8, 32	sylib 218	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
34		unixp 6238	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
35	33, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
36	35	fveq2d 6836	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
37		rankuni 9773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ (rank‘∪ (𝐴 × 𝐵))
38		rankuni 9773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (rank‘∪ (𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵))
39	38	unieqi 4873	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∪ (rank‘∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
40	37, 39	eqtri 2757	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
41	36, 40	eqtr3di 2784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
42		eqimss 3990	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)) → (rank‘(𝐴 ∪ 𝐵)) ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
43	41, 42	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (rank‘(𝐴 ∪ 𝐵)) ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
44	43	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → (rank‘(𝐴 ∪ 𝐵)) ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
45	27, 44	eqsstrrd 3967	. . . . . . . . . . . . . . . . 17 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → suc 𝑥 ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
46	45	adantrr 717	. . . . . . . . . . . . . . . 16 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc 𝑥 ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
47		limuni 6377	. . . . . . . . . . . . . . . . 17 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ∪ (rank‘(𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
48	47	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → ∪ (rank‘(𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
49	46, 48	sseqtrrd 3969	. . . . . . . . . . . . . . 15 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc 𝑥 ⊆ ∪ (rank‘(𝐴 × 𝐵)))
50		vex 3442	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
51		rankon 9705	. . . . . . . . . . . . . . . . . 18 ⊢ (rank‘(𝐴 × 𝐵)) ∈ On
52	51	onordi 6428	. . . . . . . . . . . . . . . . 17 ⊢ Ord (rank‘(𝐴 × 𝐵))
53		orduni 7732	. . . . . . . . . . . . . . . . 17 ⊢ (Ord (rank‘(𝐴 × 𝐵)) → Ord ∪ (rank‘(𝐴 × 𝐵)))
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ Ord ∪ (rank‘(𝐴 × 𝐵))
55		ordelsuc 7760	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ V ∧ Ord ∪ (rank‘(𝐴 × 𝐵))) → (𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc 𝑥 ⊆ ∪ (rank‘(𝐴 × 𝐵))))
56	50, 54, 55	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc 𝑥 ⊆ ∪ (rank‘(𝐴 × 𝐵)))
57	49, 56	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → 𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)))
58		limsuc 7789	. . . . . . . . . . . . . . 15 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc 𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵))))
59	58	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → (𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc 𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵))))
60	57, 59	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc 𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)))
61	26, 60	eqeltrd 2834	. . . . . . . . . . . 12 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)))
62		limsuc 7789	. . . . . . . . . . . . 13 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ((rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵))))
63	62	adantr 480	. . . . . . . . . . . 12 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → ((rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵))))
64	61, 63	mpbid 232	. . . . . . . . . . 11 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)))
65		ordsucelsuc 7762	. . . . . . . . . . . 12 ⊢ (Ord ∪ (rank‘(𝐴 × 𝐵)) → (suc (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc ∪ (rank‘(𝐴 × 𝐵))))
66	54, 65	ax-mp 5	. . . . . . . . . . 11 ⊢ (suc (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc ∪ (rank‘(𝐴 × 𝐵)))
67	64, 66	sylib 218	. . . . . . . . . 10 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ suc ∪ (rank‘(𝐴 × 𝐵)))
68		onsucuni2 7774	. . . . . . . . . . . 12 ⊢ (((rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦) → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
69	51, 68	mpan 690	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑦 → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
70	69	ad2antll 729	. . . . . . . . . 10 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
71	67, 70	eleqtrd 2836	. . . . . . . . 9 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐴 × 𝐵)))
72	11, 51	onsucssi 7781	. . . . . . . . 9 ⊢ (suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐴 × 𝐵)) ↔ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
73	71, 72	sylib 218	. . . . . . . 8 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
74	73	ex 412	. . . . . . 7 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))))
75	74	a1d 25	. . . . . 6 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))))
76	75	rexlimdvv 3190	. . . . 5 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (∃𝑥 ∈ On ∃𝑦 ∈ On ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))))
77	25, 76	biimtrrid 243	. . . 4 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ((∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦) → suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))))
78	24, 77	mtoi 199	. . 3 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦))
79		ianor 983	. . . . . 6 ⊢ (¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦) ↔ (¬ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∨ ¬ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦))
80		un00 4395	. . . . . . . . . . . . . 14 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅)
81		animorl 979	. . . . . . . . . . . . . 14 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
82	80, 81	sylbir 235	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅))
83		xpeq0 6116	. . . . . . . . . . . . 13 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅))
84	82, 83	sylibr 234	. . . . . . . . . . . 12 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
85	84	con3i 154	. . . . . . . . . . 11 ⊢ (¬ (𝐴 × 𝐵) = ∅ → ¬ (𝐴 ∪ 𝐵) = ∅)
86	31, 85	sylbir 235	. . . . . . . . . 10 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → ¬ (𝐴 ∪ 𝐵) = ∅)
87	19, 20	unex 7687	. . . . . . . . . . . 12 ⊢ (𝐴 ∪ 𝐵) ∈ V
88	87	rankeq0 9771	. . . . . . . . . . 11 ⊢ ((𝐴 ∪ 𝐵) = ∅ ↔ (rank‘(𝐴 ∪ 𝐵)) = ∅)
89	88	notbii 320	. . . . . . . . . 10 ⊢ (¬ (𝐴 ∪ 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 ∪ 𝐵)) = ∅)
90	86, 89	sylib 218	. . . . . . . . 9 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → ¬ (rank‘(𝐴 ∪ 𝐵)) = ∅)
91	9	onordi 6428	. . . . . . . . . . 11 ⊢ Ord (rank‘(𝐴 ∪ 𝐵))
92		ordzsl 7785	. . . . . . . . . . 11 ⊢ (Ord (rank‘(𝐴 ∪ 𝐵)) ↔ ((rank‘(𝐴 ∪ 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 ∪ 𝐵))))
93	91, 92	mpbi 230	. . . . . . . . . 10 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 ∪ 𝐵)))
94	93	3ori 1426	. . . . . . . . 9 ⊢ ((¬ (rank‘(𝐴 ∪ 𝐵)) = ∅ ∧ ¬ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → Lim (rank‘(𝐴 ∪ 𝐵)))
95	90, 94	sylan 580	. . . . . . . 8 ⊢ ((¬ (rank‘(𝐴 × 𝐵)) = ∅ ∧ ¬ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → Lim (rank‘(𝐴 ∪ 𝐵)))
96	95	ex 412	. . . . . . 7 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (¬ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 → Lim (rank‘(𝐴 ∪ 𝐵))))
97		ordzsl 7785	. . . . . . . . . 10 ⊢ (Ord (rank‘(𝐴 × 𝐵)) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦 ∨ Lim (rank‘(𝐴 × 𝐵))))
98	52, 97	mpbi 230	. . . . . . . . 9 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦 ∨ Lim (rank‘(𝐴 × 𝐵)))
99	98	3ori 1426	. . . . . . . 8 ⊢ ((¬ (rank‘(𝐴 × 𝐵)) = ∅ ∧ ¬ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦) → Lim (rank‘(𝐴 × 𝐵)))
100	99	ex 412	. . . . . . 7 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (¬ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦 → Lim (rank‘(𝐴 × 𝐵))))
101	96, 100	orim12d 966	. . . . . 6 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → ((¬ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∨ ¬ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦) → (Lim (rank‘(𝐴 ∪ 𝐵)) ∨ Lim (rank‘(𝐴 × 𝐵)))))
102	79, 101	biimtrid 242	. . . . 5 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦) → (Lim (rank‘(𝐴 ∪ 𝐵)) ∨ Lim (rank‘(𝐴 × 𝐵)))))
103	102	imp 406	. . . 4 ⊢ ((¬ (rank‘(𝐴 × 𝐵)) = ∅ ∧ ¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → (Lim (rank‘(𝐴 ∪ 𝐵)) ∨ Lim (rank‘(𝐴 × 𝐵))))
104		simpl 482	. . . . . . . 8 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ ¬ (rank‘(𝐴 × 𝐵)) = ∅) → Lim (rank‘(𝐴 ∪ 𝐵)))
105	30	necon3abii 2976	. . . . . . . . . 10 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
106	19, 20	rankxplim 9789	. . . . . . . . . 10 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
107	105, 106	sylan2br 595	. . . . . . . . 9 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ ¬ (rank‘(𝐴 × 𝐵)) = ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
108		limeq 6327	. . . . . . . . 9 ⊢ ((rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)) → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 ∪ 𝐵))))
109	107, 108	syl 17	. . . . . . . 8 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ ¬ (rank‘(𝐴 × 𝐵)) = ∅) → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim (rank‘(𝐴 ∪ 𝐵))))
110	104, 109	mpbird 257	. . . . . . 7 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ ¬ (rank‘(𝐴 × 𝐵)) = ∅) → Lim (rank‘(𝐴 × 𝐵)))
111	110	expcom 413	. . . . . 6 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (Lim (rank‘(𝐴 ∪ 𝐵)) → Lim (rank‘(𝐴 × 𝐵))))
112		idd 24	. . . . . 6 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵))))
113	111, 112	jaod 859	. . . . 5 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → ((Lim (rank‘(𝐴 ∪ 𝐵)) ∨ Lim (rank‘(𝐴 × 𝐵))) → Lim (rank‘(𝐴 × 𝐵))))
114	113	adantr 480	. . . 4 ⊢ ((¬ (rank‘(𝐴 × 𝐵)) = ∅ ∧ ¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → ((Lim (rank‘(𝐴 ∪ 𝐵)) ∨ Lim (rank‘(𝐴 × 𝐵))) → Lim (rank‘(𝐴 × 𝐵))))
115	103, 114	mpd 15	. . 3 ⊢ ((¬ (rank‘(𝐴 × 𝐵)) = ∅ ∧ ¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → Lim (rank‘(𝐴 × 𝐵)))
116	8, 78, 115	syl2anc 584	. 2 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 × 𝐵)))
117	1, 116	impbii 209	1 ⊢ (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim ∪ (rank‘(𝐴 × 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∃wrex 3058 Vcvv 3438 ∪ cun 3897 ⊆ wss 3899 ∅c0 4283 ∪ cuni 4861 × cxp 5620 Ord word 6314 Oncon0 6315 Lim wlim 6316 suc csuc 6317 ‘cfv 6490 rankcrnk 9673

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-reg 9495 ax-inf2 9548

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7359 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-r1 9674 df-rank 9675

This theorem is referenced by: rankxpsuc 9792

W3C validator