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Theorem rankxplim3 9817

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 6379	. 2 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim ∪ (rank‘(𝐴 × 𝐵)))
2		0ellim 6380	. . . 4 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ∅ ∈ ∪ (rank‘(𝐴 × 𝐵)))
3		n0i 4293	. . . 4 ⊢ (∅ ∈ ∪ (rank‘(𝐴 × 𝐵)) → ¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
4		unieq 4876	. . . . . 6 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ → ∪ (rank‘(𝐴 × 𝐵)) = ∪ ∅)
5		uni0 4896	. . . . . 6 ⊢ ∪ ∅ = ∅
6	4, 5	eqtrdi 2792	. . . . 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 9731	. . . . . . . . . 10 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ On
10	9	onsuci 7774	. . . . . . . . 9 ⊢ suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
11	10	onsuci 7774	. . . . . . . 8 ⊢ suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
12	11	elexi 3464	. . . . . . 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 7774	. . . . . . . 8 ⊢ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ On
15		ontri1 6351	. . . . . . . 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 690	. . . . . . 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 229	. . . . 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 9812	. . . . . 6 ⊢ (rank‘(𝐴 × 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))
22		sstr 3952	. . . . . 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 689	. . . . 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 196	. . . 4 ⊢ ¬ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))
25		reeanv 3217	. . . . 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 769	. . . . . . . . . . . . 13 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥)
27		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥)
28		df-ne 2944	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (𝐴 × 𝐵) = ∅)
29	19, 20	xpex 7687	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐴 × 𝐵) ∈ V
30	29	rankeq0 9797	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
31	30	notbii 319	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ (𝐴 × 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
32	28, 31	bitr2i 275	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ ↔ (𝐴 × 𝐵) ≠ ∅)
33	8, 32	sylib 217	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (𝐴 × 𝐵) ≠ ∅)
34		unixp 6234	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
35	33, 34	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
36	35	fveq2d 6846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
37		rankuni 9799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ (rank‘∪ (𝐴 × 𝐵))
38		rankuni 9799	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (rank‘∪ (𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵))
39	38	unieqi 4878	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∪ (rank‘∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
40	37, 39	eqtri 2764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
41	36, 40	eqtr3di 2791	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
42		eqimss 4000	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → (rank‘(𝐴 ∪ 𝐵)) ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
45	27, 44	eqsstrrd 3983	. . . . . . . . . . . . . . . . 17 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥) → suc 𝑥 ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
46	45	adantrr 715	. . . . . . . . . . . . . . . 16 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc 𝑥 ⊆ ∪ ∪ (rank‘(𝐴 × 𝐵)))
47		limuni 6378	. . . . . . . . . . . . . . . . 17 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ∪ (rank‘(𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
48	47	adantr 481	. . . . . . . . . . . . . . . 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 3985	. . . . . . . . . . . . . . 15 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc 𝑥 ⊆ ∪ (rank‘(𝐴 × 𝐵)))
50		vex 3449	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
51		rankon 9731	. . . . . . . . . . . . . . . . . 18 ⊢ (rank‘(𝐴 × 𝐵)) ∈ On
52	51	onordi 6428	. . . . . . . . . . . . . . . . 17 ⊢ Ord (rank‘(𝐴 × 𝐵))
53		orduni 7724	. . . . . . . . . . . . . . . . 17 ⊢ (Ord (rank‘(𝐴 × 𝐵)) → Ord ∪ (rank‘(𝐴 × 𝐵)))
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ Ord ∪ (rank‘(𝐴 × 𝐵))
55		ordelsuc 7755	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ V ∧ Ord ∪ (rank‘(𝐴 × 𝐵))) → (𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc 𝑥 ⊆ ∪ (rank‘(𝐴 × 𝐵))))
56	50, 54, 55	mp2an 690	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc 𝑥 ⊆ ∪ (rank‘(𝐴 × 𝐵)))
57	49, 56	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → 𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)))
58		limsuc 7785	. . . . . . . . . . . . . . 15 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → (𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc 𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵))))
59	58	adantr 481	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 13 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc 𝑥 ∈ ∪ (rank‘(𝐴 × 𝐵)))
61	26, 60	eqeltrd 2838	. . . . . . . . . . . 12 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)))
62		limsuc 7785	. . . . . . . . . . . . 13 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ((rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)) ↔ suc (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵))))
63	62	adantr 481	. . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ ((Lim ∪ (rank‘(𝐴 × 𝐵)) ∧ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦)) → suc (rank‘(𝐴 ∪ 𝐵)) ∈ ∪ (rank‘(𝐴 × 𝐵)))
65		ordsucelsuc 7757	. . . . . . . . . . . 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 217	. . . . . . . . . 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 7769	. . . . . . . . . . . 12 ⊢ (((rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑦) → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
69	51, 68	mpan 688	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑦 → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
70	69	ad2antll 727	. . . . . . . . . 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 2840	. . . . . . . . 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 7777	. . . . . . . . 9 ⊢ (suc suc (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐴 × 𝐵)) ↔ suc suc suc (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))
73	71, 72	sylib 217	. . . . . . . 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 413	. . . . . . 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 3204	. . . . 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 242	. . . 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 198	. . 3 ⊢ (Lim ∪ (rank‘(𝐴 × 𝐵)) → ¬ (∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∧ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦))
79		ianor 980	. . . . . 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 4402	. . . . . . . . . . . . . 14 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴 ∪ 𝐵) = ∅)
81		animorl 976	. . . . . . . . . . . . . 14 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
82	80, 81	sylbir 234	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅))
83		xpeq0 6112	. . . . . . . . . . . . 13 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅))
84	82, 83	sylibr 233	. . . . . . . . . . . 12 ⊢ ((𝐴 ∪ 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
85	84	con3i 154	. . . . . . . . . . 11 ⊢ (¬ (𝐴 × 𝐵) = ∅ → ¬ (𝐴 ∪ 𝐵) = ∅)
86	31, 85	sylbir 234	. . . . . . . . . 10 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → ¬ (𝐴 ∪ 𝐵) = ∅)
87	19, 20	unex 7680	. . . . . . . . . . . 12 ⊢ (𝐴 ∪ 𝐵) ∈ V
88	87	rankeq0 9797	. . . . . . . . . . 11 ⊢ ((𝐴 ∪ 𝐵) = ∅ ↔ (rank‘(𝐴 ∪ 𝐵)) = ∅)
89	88	notbii 319	. . . . . . . . . 10 ⊢ (¬ (𝐴 ∪ 𝐵) = ∅ ↔ ¬ (rank‘(𝐴 ∪ 𝐵)) = ∅)
90	86, 89	sylib 217	. . . . . . . . 9 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → ¬ (rank‘(𝐴 ∪ 𝐵)) = ∅)
91	9	onordi 6428	. . . . . . . . . . 11 ⊢ Ord (rank‘(𝐴 ∪ 𝐵))
92		ordzsl 7781	. . . . . . . . . . 11 ⊢ (Ord (rank‘(𝐴 ∪ 𝐵)) ↔ ((rank‘(𝐴 ∪ 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 ∪ 𝐵))))
93	91, 92	mpbi 229	. . . . . . . . . 10 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 ∪ 𝐵)))
94	93	3ori 1424	. . . . . . . . 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 413	. . . . . . 7 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (¬ ∃𝑥 ∈ On (rank‘(𝐴 ∪ 𝐵)) = suc 𝑥 → Lim (rank‘(𝐴 ∪ 𝐵))))
97		ordzsl 7781	. . . . . . . . . 10 ⊢ (Ord (rank‘(𝐴 × 𝐵)) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦 ∨ Lim (rank‘(𝐴 × 𝐵))))
98	52, 97	mpbi 229	. . . . . . . . 9 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦 ∨ Lim (rank‘(𝐴 × 𝐵)))
99	98	3ori 1424	. . . . . . . 8 ⊢ ((¬ (rank‘(𝐴 × 𝐵)) = ∅ ∧ ¬ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦) → Lim (rank‘(𝐴 × 𝐵)))
100	99	ex 413	. . . . . . 7 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (¬ ∃𝑦 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑦 → Lim (rank‘(𝐴 × 𝐵))))
101	96, 100	orim12d 963	. . . . . 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 241	. . . . 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 407	. . . 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 483	. . . . . . . 8 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ ¬ (rank‘(𝐴 × 𝐵)) = ∅) → Lim (rank‘(𝐴 ∪ 𝐵)))
105	30	necon3abii 2990	. . . . . . . . . 10 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
106	19, 20	rankxplim 9815	. . . . . . . . . 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 6329	. . . . . . . . 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 256	. . . . . . 7 ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ ¬ (rank‘(𝐴 × 𝐵)) = ∅) → Lim (rank‘(𝐴 × 𝐵)))
111	110	expcom 414	. . . . . 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 857	. . . . 5 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → ((Lim (rank‘(𝐴 ∪ 𝐵)) ∨ Lim (rank‘(𝐴 × 𝐵))) → Lim (rank‘(𝐴 × 𝐵))))
114	113	adantr 481	. . . 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 208	1 ⊢ (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim ∪ (rank‘(𝐴 × 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 Vcvv 3445 ∪ cun 3908 ⊆ wss 3910 ∅c0 4282 ∪ cuni 4865 × cxp 5631 Ord word 6316 Oncon0 6317 Lim wlim 6318 suc csuc 6319 ‘cfv 6496 rankcrnk 9699

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-reg 9528 ax-inf2 9577

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-r1 9700 df-rank 9701

This theorem is referenced by: rankxpsuc 9818

W3C validator