Theorem rankxpsuc 9901

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 9898 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 6276	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
2	1	fveq2d 6885	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
3		rankuni 9882	. . . . . . . 8 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ (rank‘∪ (𝐴 × 𝐵))
4		rankuni 9882	. . . . . . . . 9 ⊢ (rank‘∪ (𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵))
5	4	unieqi 4900	. . . . . . . 8 ⊢ ∪ (rank‘∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
6	3, 5	eqtri 2759	. . . . . . 7 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
7	2, 6	eqtr3di 2786	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
8		suc11reg 9638	. . . . . 6 ⊢ (suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ ∪ (rank‘(𝐴 × 𝐵)) ↔ (rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
9	7, 8	sylibr 234	. . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ → suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ ∪ (rank‘(𝐴 × 𝐵)))
10	9	adantl 481	. . . 4 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ ∪ (rank‘(𝐴 × 𝐵)))
11		fvex 6894	. . . . . . . . . . . . . 14 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ V
12		eleq1 2823	. . . . . . . . . . . . . 14 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ((rank‘(𝐴 ∪ 𝐵)) ∈ V ↔ suc 𝐶 ∈ V))
13	11, 12	mpbii 233	. . . . . . . . . . . . 13 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → suc 𝐶 ∈ V)
14		sucexb 7803	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ V ↔ suc 𝐶 ∈ V)
15	13, 14	sylibr 234	. . . . . . . . . . . 12 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → 𝐶 ∈ V)
16		nlimsucg 7842	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ V → ¬ Lim suc 𝐶)
17	15, 16	syl 17	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ¬ Lim suc 𝐶)
18		limeq 6369	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → (Lim (rank‘(𝐴 ∪ 𝐵)) ↔ Lim suc 𝐶))
19	17, 18	mtbird 325	. . . . . . . . . 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 9899	. . . . . . . . . 10 ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵)))
23	19, 22	nsyl 140	. . . . . . . . 9 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ¬ Lim (rank‘(𝐴 × 𝐵)))
24	20, 21	xpex 7752	. . . . . . . . . . . . . 14 ⊢ (𝐴 × 𝐵) ∈ V
25	24	rankeq0 9880	. . . . . . . . . . . . 13 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
26	25	necon3abii 2979	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
27		rankon 9814	. . . . . . . . . . . . . . . 16 ⊢ (rank‘(𝐴 × 𝐵)) ∈ On
28	27	onordi 6470	. . . . . . . . . . . . . . 15 ⊢ Ord (rank‘(𝐴 × 𝐵))
29		ordzsl 7845	. . . . . . . . . . . . . . 15 ⊢ (Ord (rank‘(𝐴 × 𝐵)) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
30	28, 29	mpbi 230	. . . . . . . . . . . . . 14 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))
31		3orass 1089	. . . . . . . . . . . . . 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 230	. . . . . . . . . . . . 13 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
33	32	ori 861	. . . . . . . . . . . 12 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
34	26, 33	sylbi 217	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
35	34	ord 864	. . . . . . . . . 10 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (¬ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → Lim (rank‘(𝐴 × 𝐵))))
36	35	con1d 145	. . . . . . . . 9 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (¬ Lim (rank‘(𝐴 × 𝐵)) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
37	23, 36	syl5com 31	. . . . . . . 8 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ((𝐴 × 𝐵) ≠ ∅ → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥))
38		nlimsucg 7842	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → ¬ Lim suc 𝑥)
39	38	elv 3469	. . . . . . . . . . 11 ⊢ ¬ Lim suc 𝑥
40		limeq 6369	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim suc 𝑥))
41	39, 40	mtbiri 327	. . . . . . . . . 10 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
42	41	rexlimivw 3138	. . . . . . . . 9 ⊢ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
43	20, 21	rankxplim3 9900	. . . . . . . . 9 ⊢ (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim ∪ (rank‘(𝐴 × 𝐵)))
44	42, 43	sylnib 328	. . . . . . . 8 ⊢ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim ∪ (rank‘(𝐴 × 𝐵)))
45	37, 44	syl6com 37	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ¬ Lim ∪ (rank‘(𝐴 × 𝐵))))
46		unixp0 6277	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅)
47	24	uniex 7740	. . . . . . . . . . . . 13 ⊢ ∪ (𝐴 × 𝐵) ∈ V
48	47	rankeq0 9880	. . . . . . . . . . . 12 ⊢ (∪ (𝐴 × 𝐵) = ∅ ↔ (rank‘∪ (𝐴 × 𝐵)) = ∅)
49	4	eqeq1i 2741	. . . . . . . . . . . 12 ⊢ ((rank‘∪ (𝐴 × 𝐵)) = ∅ ↔ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
50	46, 48, 49	3bitri 297	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
51	50	necon3abii 2979	. . . . . . . . . 10 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
52		onuni 7787	. . . . . . . . . . . . . . 15 ⊢ ((rank‘(𝐴 × 𝐵)) ∈ On → ∪ (rank‘(𝐴 × 𝐵)) ∈ On)
53	27, 52	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ∪ (rank‘(𝐴 × 𝐵)) ∈ On
54	53	onordi 6470	. . . . . . . . . . . . 13 ⊢ Ord ∪ (rank‘(𝐴 × 𝐵))
55		ordzsl 7845	. . . . . . . . . . . . 13 ⊢ (Ord ∪ (rank‘(𝐴 × 𝐵)) ↔ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
56	54, 55	mpbi 230	. . . . . . . . . . . 12 ⊢ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵)))
57		3orass 1089	. . . . . . . . . . . 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 230	. . . . . . . . . . 11 ⊢ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
59	58	ori 861	. . . . . . . . . 10 ⊢ (¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
60	51, 59	sylbi 217	. . . . . . . . 9 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
61	60	ord 864	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (¬ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 → Lim ∪ (rank‘(𝐴 × 𝐵))))
62	61	con1d 145	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (¬ Lim ∪ (rank‘(𝐴 × 𝐵)) → ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥))
63	45, 62	syld 47	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥))
64	63	impcom 407	. . . . 5 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥)
65		onsucuni2 7833	. . . . . . 7 ⊢ ((∪ (rank‘(𝐴 × 𝐵)) ∈ On ∧ ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
66	53, 65	mpan 690	. . . . . 6 ⊢ (∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
67	66	rexlimivw 3138	. . . . 5 ⊢ (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
68	64, 67	syl 17	. . . 4 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
69	10, 68	eqtrd 2771	. . 3 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴 ∪ 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
70		suc11reg 9638	. . 3 ⊢ (suc suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ (rank‘(𝐴 × 𝐵)) ↔ suc (rank‘(𝐴 ∪ 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
71	69, 70	sylibr 234	. 2 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ (rank‘(𝐴 × 𝐵)))
72	37	imp 406	. . 3 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥)
73		onsucuni2 7833	. . . . 5 ⊢ (((rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
74	27, 73	mpan 690	. . . 4 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
75	74	rexlimivw 3138	. . 3 ⊢ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
76	72, 75	syl 17	. 2 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
77	71, 76	eqtr2d 2772	1 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∃wrex 3061  Vcvv 3464   ∪ cun 3929  ∅c0 4313  ∪ cuni 4888   × cxp 5657  Ord word 6356  Oncon0 6357  Lim wlim 6358  suc csuc 6359  ‘cfv 6536  rankcrnk 9782

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-reg 9611  ax-inf2 9660

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-r1 9783  df-rank 9784

This theorem is referenced by: (None)