Theorem rankxpsuc 9830

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 9827 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 6258	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))
2	1	fveq2d 6860	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘∪ ∪ (𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))
3		rankuni 9811	. . . . . . . 8 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ (rank‘∪ (𝐴 × 𝐵))
4		rankuni 9811	. . . . . . . . 9 ⊢ (rank‘∪ (𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵))
5	4	unieqi 4871	. . . . . . . 8 ⊢ ∪ (rank‘∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
6	3, 5	eqtri 2779	. . . . . . 7 ⊢ (rank‘∪ ∪ (𝐴 × 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵))
7	2, 6	eqtr3di 2806	. . . . . 6 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
8		suc11reg 9564	. . . . . 6 ⊢ (suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ ∪ (rank‘(𝐴 × 𝐵)) ↔ (rank‘(𝐴 ∪ 𝐵)) = ∪ ∪ (rank‘(𝐴 × 𝐵)))
9	7, 8	sylibr 236	. . . . 5 ⊢ ((𝐴 × 𝐵) ≠ ∅ → suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ ∪ (rank‘(𝐴 × 𝐵)))
10	9	adantl 484	. . . 4 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ ∪ (rank‘(𝐴 × 𝐵)))
11		fvex 6869	. . . . . . . . . . . . . 14 ⊢ (rank‘(𝐴 ∪ 𝐵)) ∈ V
12		eleq1 2844	. . . . . . . . . . . . . 14 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ((rank‘(𝐴 ∪ 𝐵)) ∈ V ↔ suc 𝐶 ∈ V))
13	11, 12	mpbii 235	. . . . . . . . . . . . 13 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → suc 𝐶 ∈ V)
14		sucexb 7776	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ V ↔ suc 𝐶 ∈ V)
15	13, 14	sylibr 236	. . . . . . . . . . . 12 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → 𝐶 ∈ V)
16		nlimsucg 7811	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ V → ¬ Lim suc 𝐶)
17	15, 16	syl 17	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → ¬ Lim suc 𝐶)
18		limeq 6347	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 → (Lim (rank‘(𝐴 ∪ 𝐵)) ↔ Lim suc 𝐶))
19	17, 18	mtbird 327	. . . . . . . . . 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 9828	. . . . . . . . . 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 7725	. . . . . . . . . . . . . 14 ⊢ (𝐴 × 𝐵) ∈ V
25	24	rankeq0 9809	. . . . . . . . . . . . 13 ⊢ ((𝐴 × 𝐵) = ∅ ↔ (rank‘(𝐴 × 𝐵)) = ∅)
26	25	necon3abii 2997	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ (rank‘(𝐴 × 𝐵)) = ∅)
27		rankon 9743	. . . . . . . . . . . . . . . 16 ⊢ (rank‘(𝐴 × 𝐵)) ∈ On
28	27	onordi 6448	. . . . . . . . . . . . . . 15 ⊢ Ord (rank‘(𝐴 × 𝐵))
29		ordzsl 7814	. . . . . . . . . . . . . . 15 ⊢ (Ord (rank‘(𝐴 × 𝐵)) ↔ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
30	28, 29	mpbi 232	. . . . . . . . . . . . . 14 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵)))
31		3orass 1098	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 13 ⊢ ((rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
33	32	ori 870	. . . . . . . . . . . 12 ⊢ (¬ (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
34	26, 33	sylbi 219	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim (rank‘(𝐴 × 𝐵))))
35	34	ord 873	. . . . . . . . . 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 7811	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → ¬ Lim suc 𝑥)
39	38	elv 3453	. . . . . . . . . . 11 ⊢ ¬ Lim suc 𝑥
40		limeq 6347	. . . . . . . . . . 11 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim suc 𝑥))
41	39, 40	mtbiri 329	. . . . . . . . . 10 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
42	41	rexlimivw 3153	. . . . . . . . 9 ⊢ (∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥 → ¬ Lim (rank‘(𝐴 × 𝐵)))
43	20, 21	rankxplim3 9829	. . . . . . . . 9 ⊢ (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim ∪ (rank‘(𝐴 × 𝐵)))
44	42, 43	sylnib 330	. . . . . . . 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 6259	. . . . . . . . . . . 12 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (𝐴 × 𝐵) = ∅)
47	24	uniex 7713	. . . . . . . . . . . . 13 ⊢ ∪ (𝐴 × 𝐵) ∈ V
48	47	rankeq0 9809	. . . . . . . . . . . 12 ⊢ (∪ (𝐴 × 𝐵) = ∅ ↔ (rank‘∪ (𝐴 × 𝐵)) = ∅)
49	4	eqeq1i 2761	. . . . . . . . . . . 12 ⊢ ((rank‘∪ (𝐴 × 𝐵)) = ∅ ↔ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
50	46, 48, 49	3bitri 299	. . . . . . . . . . 11 ⊢ ((𝐴 × 𝐵) = ∅ ↔ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
51	50	necon3abii 2997	. . . . . . . . . 10 ⊢ ((𝐴 × 𝐵) ≠ ∅ ↔ ¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅)
52		onuni 7760	. . . . . . . . . . . . . . 15 ⊢ ((rank‘(𝐴 × 𝐵)) ∈ On → ∪ (rank‘(𝐴 × 𝐵)) ∈ On)
53	27, 52	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ∪ (rank‘(𝐴 × 𝐵)) ∈ On
54	53	onordi 6448	. . . . . . . . . . . . 13 ⊢ Ord ∪ (rank‘(𝐴 × 𝐵))
55		ordzsl 7814	. . . . . . . . . . . . 13 ⊢ (Ord ∪ (rank‘(𝐴 × 𝐵)) ↔ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
56	54, 55	mpbi 232	. . . . . . . . . . . 12 ⊢ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵)))
57		3orass 1098	. . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ (∪ (rank‘(𝐴 × 𝐵)) = ∅ ∨ (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
59	58	ori 870	. . . . . . . . . 10 ⊢ (¬ ∪ (rank‘(𝐴 × 𝐵)) = ∅ → (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
60	51, 59	sylbi 219	. . . . . . . . 9 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 ∨ Lim ∪ (rank‘(𝐴 × 𝐵))))
61	60	ord 873	. . . . . . . 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 410	. . . . 5 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥)
65		onsucuni2 7803	. . . . . . 7 ⊢ ((∪ (rank‘(𝐴 × 𝐵)) ∈ On ∧ ∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
66	53, 65	mpan 698	. . . . . 6 ⊢ (∪ (rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ ∪ (rank‘(𝐴 × 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
67	66	rexlimivw 3153	. . . . 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 2791	. . 3 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc (rank‘(𝐴 ∪ 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
70		suc11reg 9564	. . 3 ⊢ (suc suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ (rank‘(𝐴 × 𝐵)) ↔ suc (rank‘(𝐴 ∪ 𝐵)) = ∪ (rank‘(𝐴 × 𝐵)))
71	69, 70	sylibr 236	. 2 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → suc suc (rank‘(𝐴 ∪ 𝐵)) = suc ∪ (rank‘(𝐴 × 𝐵)))
72	37	imp 409	. . 3 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → ∃𝑥 ∈ On (rank‘(𝐴 × 𝐵)) = suc 𝑥)
73		onsucuni2 7803	. . . . 5 ⊢ (((rank‘(𝐴 × 𝐵)) ∈ On ∧ (rank‘(𝐴 × 𝐵)) = suc 𝑥) → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
74	27, 73	mpan 698	. . . 4 ⊢ ((rank‘(𝐴 × 𝐵)) = suc 𝑥 → suc ∪ (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 × 𝐵)))
75	74	rexlimivw 3153	. . 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 2792	1 ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 856   ∨ w3o 1094   = wceq 1554   ∈ wcel 2136   ≠ wne 2951  ∃wrex 3080  Vcvv 3448   ∪ cun 3897  ∅c0 4280  ∪ cuni 4859   × cxp 5638  Ord word 6334  Oncon0 6335  Lim wlim 6336  suc csuc 6337  ‘cfv 6510  rankcrnk 9711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707  ax-reg 9530  ax-inf2 9586

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4900  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-om 7836  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-r1 9712  df-rank 9713

This theorem is referenced by: (None)