Theorem rankxpsuc 9842

Description: The rank of a Cartesian product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 9839 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.)

Hypotheses

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

Assertion

Ref	Expression
rankxpsuc	⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵)))

Proof of Theorem rankxpsuc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unixp 6273	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
2	1	fveq2d 6875	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
3		rankuni 9823	. . . . . . . 8 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ (rank‘∪ (𝐴 × 𝐵))
4		rankuni 9823	. . . . . . . . 9 ⊢ (rank‘∪ (𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵))
5	4	unieqi 4880	. . . . . . . 8 ⊢ ∪ (rank‘∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
6	3, 5	eqtri 2788	. . . . . . 7 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
7	2, 6	eqtr3di 2815	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
8		suc11reg 9576	. . . . . 6 ⊢ (suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ ∪ (rank‘(𝐴 × 𝐵)) ↔ (rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
9	7, 8	sylibr 237	. . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ → suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ ∪ (rank‘(𝐴 × 𝐵)))
10	9	adantl 486	. . . 4 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ ∪ (rank‘(𝐴 × 𝐵)))
11		fvex 6884	. . . . . . . . . . . . . 14 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ V
12		eleq1 2853	. . . . . . . . . . . . . 14 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ((rank‘(𝐴 ∪ 𝐵)) ∈ V ↔ suc 𝐶 ∈ V))
13	11, 12	mpbii 236	. . . . . . . . . . . . 13 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → suc 𝐶 ∈ V)
14		sucexb 7791	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ V ↔ suc 𝐶 ∈ V)
15	13, 14	sylibr 237	. . . . . . . . . . . 12 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → 𝐶 ∈ V)
16		nlimsucg 7826	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ V → ¬ Lim suc 𝐶)
17	15, 16	syl 18	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ¬ Lim suc 𝐶)
18		limeq 6362	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → (Lim (rank‘(𝐴 ∪ 𝐵)) ↔ Lim suc 𝐶))
19	17, 18	mtbird 328	. . . . . . . . . 10 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴 ∪ 𝐵)))
20		rankxplim.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
21		rankxplim.2	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
22	20, 21	rankxplim2 9840	. . . . . . . . . 10 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵)))
23	19, 22	nsyl 141	. . . . . . . . 9 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴 × 𝐵)))
24	20, 21	xpex 7740	. . . . . . . . . . . . . 14 ⊢ (𝐴 × 𝐵) ∈ V
25	24	rankeq0 9821	. . . . . . . . . . . . 13 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
26	25	necon3abii 3006	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
27		rankon 9755	. . . . . . . . . . . . . . . 16 ⊢ (rank‘(𝐴 × 𝐵)) ∈ On
28	27	onordi 6463	. . . . . . . . . . . . . . 15 ⊢ Ord (rank‘(𝐴 × 𝐵))
29		ordzsl 7829	. . . . . . . . . . . . . . 15 ⊢ (Ord (rank‘(𝐴 × 𝐵)) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
30	28, 29	mpbi 233	. . . . . . . . . . . . . 14 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))
31		3orass 1104	. . . . . . . . . . . . . 14 ⊢ (((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))))
32	30, 31	mpbi 233	. . . . . . . . . . . . 13 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
33	32	ori 874	. . . . . . . . . . . 12 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
34	26, 33	sylbi 220	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
35	34	ord 877	. . . . . . . . . 10 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (¬ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → Lim (rank‘(𝐴 × 𝐵))))
36	35	con1d 146	. . . . . . . . 9 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (¬ Lim (rank‘(𝐴 × 𝐵)) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
37	23, 36	syl5com 32	. . . . . . . 8 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ((𝐴 × 𝐵) ≠ ∅ → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
38		nlimsucg 7826	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → ¬ Lim suc 𝑥)
39	38	elv 3462	. . . . . . . . . . 11 ⊢ ¬ Lim suc 𝑥
40		limeq 6362	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim suc 𝑥))
41	39, 40	mtbiri 330	. . . . . . . . . 10 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
42	41	rexlimivw 3162	. . . . . . . . 9 ⊢ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
43	20, 21	rankxplim3 9841	. . . . . . . . 9 ⊢ (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim ∪ (rank‘(𝐴 × 𝐵)))
44	42, 43	sylnib 331	. . . . . . . 8 ⊢ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim ∪ (rank‘(𝐴 × 𝐵)))
45	37, 44	syl6com 38	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ¬ Lim ∪ (rank‘(𝐴 × 𝐵))))
46		unixp0 6274	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅)
47	24	uniex 7728	. . . . . . . . . . . . 13 ⊢ ∪ (𝐴 × 𝐵) ∈ V
48	47	rankeq0 9821	. . . . . . . . . . . 12 ⊢ (∪ (𝐴 × 𝐵) = ∅ ↔ (rank‘∪ (𝐴 × 𝐵)) = ∅)
49	4	eqeq1i 2770	. . . . . . . . . . . 12 ⊢ ((rank‘∪ (𝐴 × 𝐵)) = ∅ ↔ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
50	46, 48, 49	3bitri 300	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
51	50	necon3abii 3006	. . . . . . . . . 10 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
52		onuni 7775	. . . . . . . . . . . . . . 15 ⊢ ((rank‘(𝐴 × 𝐵)) ∈ On → ∪ (rank‘(𝐴 × 𝐵)) ∈ On)
53	27, 52	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ∪ (rank‘(𝐴 × 𝐵)) ∈ On
54	53	onordi 6463	. . . . . . . . . . . . 13 ⊢ Ord ∪ (rank‘(𝐴 × 𝐵))
55		ordzsl 7829	. . . . . . . . . . . . 13 ⊢ (Ord ∪ (rank‘(𝐴 × 𝐵)) ↔ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
56	54, 55	mpbi 233	. . . . . . . . . . . 12 ⊢ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵)))
57		3orass 1104	. . . . . . . . . . . 12 ⊢ ((∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))) ↔ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵)))))
58	56, 57	mpbi 233	. . . . . . . . . . 11 ⊢ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
59	58	ori 874	. . . . . . . . . 10 ⊢ (¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
60	51, 59	sylbi 220	. . . . . . . . 9 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
61	60	ord 877	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (¬ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 → Lim ∪ (rank‘(𝐴 × 𝐵))))
62	61	con1d 146	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (¬ Lim ∪ (rank‘(𝐴 × 𝐵)) → ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥))
63	45, 62	syld 48	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥))
64	63	impcom 412	. . . . 5 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥)
65		onsucuni2 7818	. . . . . . 7 ⊢ ((∪ (rank‘(𝐴 × 𝐵)) ∈ On ∧ ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
66	53, 65	mpan 702	. . . . . 6 ⊢ (∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
67	66	rexlimivw 3162	. . . . 5 ⊢ (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
68	64, 67	syl 18	. . . 4 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
69	10, 68	eqtrd 2800	. . 3 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴 ∪ 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
70		suc11reg 9576	. . 3 ⊢ (suc suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ (rank‘(𝐴 × 𝐵)) ↔ suc (rank‘(𝐴 ∪ 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
71	69, 70	sylibr 237	. 2 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ (rank‘(𝐴 × 𝐵)))
72	37	imp 411	. . 3 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥)
73		onsucuni2 7818	. . . . 5 ⊢ (((rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
74	27, 73	mpan 702	. . . 4 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
75	74	rexlimivw 3162	. . 3 ⊢ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
76	72, 75	syl 18	. 2 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
77	71, 76	eqtr2d 2801	1 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   ∨ w3o 1100   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089  Vcvv 3457   ∪ cun 3905  ∅c0 4288  ∪ cuni 4868   × cxp 5650  Ord word 6349  Oncon0 6350  Lim wlim 6351  suc csuc 6352  ‘cfv 6525  rankcrnk 9723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-reg 9542  ax-inf2 9598

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-r1 9724  df-rank 9725

This theorem is referenced by: (None)