Theorem rankxplim 4774

Description: The rank of a cross product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 4777 for the successor case.

Hypotheses

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

Assertion

Ref	Expression
rankxplim	⊢ ((Lim (rank ‘(A ∪ B)) ⋀ (A × B) ≠ ∅) → (rank ‘(A × B)) = (rank ‘(A ∪ B)))

Proof of Theorem rankxplim

Step	Hyp	Ref	Expression
1		visset 1860	. . . . . . . . . . . 12 ⊢ x ∈ V
2		visset 1860	. . . . . . . . . . . 12 ⊢ y ∈ V
3		rankxplim.1	. . . . . . . . . . . 12 ⊢ A ∈ V
4		rankxplim.2	. . . . . . . . . . . 12 ⊢ B ∈ V
5	1, 2, 3, 4	rankelun 4769	. . . . . . . . . . 11 ⊢ (((rank ‘x) ∈ (rank ‘A) ⋀ (rank ‘y) ∈ (rank ‘B)) → (rank ‘(x ∪ y)) ∈ (rank ‘(A ∪ B)))
6	3	rankel 4742	. . . . . . . . . . 11 ⊢ (x ∈ A → (rank ‘x) ∈ (rank ‘A))
7	4	rankel 4742	. . . . . . . . . . 11 ⊢ (y ∈ B → (rank ‘y) ∈ (rank ‘B))
8	5, 6, 7	syl2an 465	. . . . . . . . . 10 ⊢ ((x ∈ A ⋀ y ∈ B) → (rank ‘(x ∪ y)) ∈ (rank ‘(A ∪ B)))
9	8	adantl 397	. . . . . . . . 9 ⊢ ((Lim (rank ‘(A ∪ B)) ⋀ (x ∈ A ⋀ y ∈ B)) → (rank ‘(x ∪ y)) ∈ (rank ‘(A ∪ B)))
10		ranklim 4747	. . . . . . . . . . 11 ⊢ (Lim (rank ‘(A ∪ B)) → ((rank ‘(x ∪ y)) ∈ (rank ‘(A ∪ B)) ↔ (rank ‘℘(x ∪ y)) ∈ (rank ‘(A ∪ B))))
11		ranklim 4747	. . . . . . . . . . 11 ⊢ (Lim (rank ‘(A ∪ B)) → ((rank ‘℘(x ∪ y)) ∈ (rank ‘(A ∪ B)) ↔ (rank ‘℘℘(x ∪ y)) ∈ (rank ‘(A ∪ B))))
12	10, 11	bitrd 539	. . . . . . . . . 10 ⊢ (Lim (rank ‘(A ∪ B)) → ((rank ‘(x ∪ y)) ∈ (rank ‘(A ∪ B)) ↔ (rank ‘℘℘(x ∪ y)) ∈ (rank ‘(A ∪ B))))
13	12	adantr 398	. . . . . . . . 9 ⊢ ((Lim (rank ‘(A ∪ B)) ⋀ (x ∈ A ⋀ y ∈ B)) → ((rank ‘(x ∪ y)) ∈ (rank ‘(A ∪ B)) ↔ (rank ‘℘℘(x ∪ y)) ∈ (rank ‘(A ∪ B))))
14	9, 13	mpbid 202	. . . . . . . 8 ⊢ ((Lim (rank ‘(A ∪ B)) ⋀ (x ∈ A ⋀ y ∈ B)) → (rank ‘℘℘(x ∪ y)) ∈ (rank ‘(A ∪ B)))
15		pwuni 2813	. . . . . . . . . . . 12 ⊢ ⟨x, y⟩ ⊆ ℘∪⟨x, y⟩
16		uniop 2864	. . . . . . . . . . . . 13 ⊢ ∪⟨x, y⟩ = {x, y}
17		pweq 2455	. . . . . . . . . . . . 13 ⊢ (∪⟨x, y⟩ = {x, y} → ℘∪⟨x, y⟩ = ℘{x, y})
18	16, 17	ax-mp 7	. . . . . . . . . . . 12 ⊢ ℘∪⟨x, y⟩ = ℘{x, y}
19	15, 18	sseqtri 2144	. . . . . . . . . . 11 ⊢ ⟨x, y⟩ ⊆ ℘{x, y}
20		pwuni 2813	. . . . . . . . . . . . 13 ⊢ {x, y} ⊆ ℘∪{x, y}
21	1, 2	unipr 2569	. . . . . . . . . . . . . 14 ⊢ ∪{x, y} = (x ∪ y)
22		pweq 2455	. . . . . . . . . . . . . 14 ⊢ (∪{x, y} = (x ∪ y) → ℘∪{x, y} = ℘(x ∪ y))
23	21, 22	ax-mp 7	. . . . . . . . . . . . 13 ⊢ ℘∪{x, y} = ℘(x ∪ y)
24	20, 23	sseqtri 2144	. . . . . . . . . . . 12 ⊢ {x, y} ⊆ ℘(x ∪ y)
25		sspwb 2811	. . . . . . . . . . . 12 ⊢ ({x, y} ⊆ ℘(x ∪ y) ↔ ℘{x, y} ⊆ ℘℘(x ∪ y))
26	24, 25	mpbi 196	. . . . . . . . . . 11 ⊢ ℘{x, y} ⊆ ℘℘(x ∪ y)
27	19, 26	sstri 2124	. . . . . . . . . 10 ⊢ ⟨x, y⟩ ⊆ ℘℘(x ∪ y)
28	1, 2	unex 2928	. . . . . . . . . . . . 13 ⊢ (x ∪ y) ∈ V
29	28	pwex 2801	. . . . . . . . . . . 12 ⊢ ℘(x ∪ y) ∈ V
30	29	pwex 2801	. . . . . . . . . . 11 ⊢ ℘℘(x ∪ y) ∈ V
31	30	rankss 4750	. . . . . . . . . 10 ⊢ (⟨x, y⟩ ⊆ ℘℘(x ∪ y) → (rank ‘⟨x, y⟩) ⊆ (rank ‘℘℘(x ∪ y)))
32	27, 31	ax-mp 7	. . . . . . . . 9 ⊢ (rank ‘⟨x, y⟩) ⊆ (rank ‘℘℘(x ∪ y))
33		rankon 4733	. . . . . . . . . 10 ⊢ (rank ‘⟨x, y⟩) ∈ On
34		rankon 4733	. . . . . . . . . 10 ⊢ (rank ‘(A ∪ B)) ∈ On
35		ontr2 3061	. . . . . . . . . 10 ⊢ (((rank ‘⟨x, y⟩) ∈ On ⋀ (rank ‘(A ∪ B)) ∈ On) → (((rank ‘⟨x, y⟩) ⊆ (rank ‘℘℘(x ∪ y)) ⋀ (rank ‘℘℘(x ∪ y)) ∈ (rank ‘(A ∪ B))) → (rank ‘⟨x, y⟩) ∈ (rank ‘(A ∪ B))))
36	33, 34, 35	mp2an 709	. . . . . . . . 9 ⊢ (((rank ‘⟨x, y⟩) ⊆ (rank ‘℘℘(x ∪ y)) ⋀ (rank ‘℘℘(x ∪ y)) ∈ (rank ‘(A ∪ B))) → (rank ‘⟨x, y⟩) ∈ (rank ‘(A ∪ B)))
37	32, 36	mpan 707	. . . . . . . 8 ⊢ ((rank ‘℘℘(x ∪ y)) ∈ (rank ‘(A ∪ B)) → (rank ‘⟨x, y⟩) ∈ (rank ‘(A ∪ B)))
38	14, 37	syl 10	. . . . . . 7 ⊢ ((Lim (rank ‘(A ∪ B)) ⋀ (x ∈ A ⋀ y ∈ B)) → (rank ‘⟨x, y⟩) ∈ (rank ‘(A ∪ B)))
39	33, 34	onsucssi 3168	. . . . . . 7 ⊢ ((rank ‘⟨x, y⟩) ∈ (rank ‘(A ∪ B)) ↔ suc (rank ‘⟨x, y⟩) ⊆ (rank ‘(A ∪ B)))
40	38, 39	sylib 205	. . . . . 6 ⊢ ((Lim (rank ‘(A ∪ B)) ⋀ (x ∈ A ⋀ y ∈ B)) → suc (rank ‘⟨x, y⟩) ⊆ (rank ‘(A ∪ B)))
41	40	ex 380	. . . . 5 ⊢ (Lim (rank ‘(A ∪ B)) → ((x ∈ A ⋀ y ∈ B) → suc (rank ‘⟨x, y⟩) ⊆ (rank ‘(A ∪ B))))
42	41	r19.21aivv 1767	. . . 4 ⊢ (Lim (rank ‘(A ∪ B)) → ∀x ∈ A ∀y ∈ B suc (rank ‘⟨x, y⟩) ⊆ (rank ‘(A ∪ B)))
43		fveq2 3781	. . . . . . . 8 ⊢ (z = ⟨x, y⟩ → (rank ‘z) = (rank ‘⟨x, y⟩))
44		suceq 3091	. . . . . . . 8 ⊢ ((rank ‘z) = (rank ‘⟨x, y⟩) → suc (rank ‘z) = suc (rank ‘⟨x, y⟩))
45	43, 44	syl 10	. . . . . . 7 ⊢ (z = ⟨x, y⟩ → suc (rank ‘z) = suc (rank ‘⟨x, y⟩))
46	45	sseq1d 2139	. . . . . 6 ⊢ (z = ⟨x, y⟩ → (suc (rank ‘z) ⊆ (rank ‘(A ∪ B)) ↔ suc (rank ‘⟨x, y⟩) ⊆ (rank ‘(A ∪ B))))
47	46	ralxp 3275	. . . . 5 ⊢ (∀z ∈ (A × B)suc (rank ‘z) ⊆ (rank ‘(A ∪ B)) ↔ ∀x ∈ A ∀y ∈ B suc (rank ‘⟨x, y⟩) ⊆ (rank ‘(A ∪ B)))
48	3, 4	xpex 3317	. . . . . 6 ⊢ (A × B) ∈ V
49	48	rankbnd 4765	. . . . 5 ⊢ (∀z ∈ (A × B)suc (rank ‘z) ⊆ (rank ‘(A ∪ B)) ↔ (rank ‘(A × B)) ⊆ (rank ‘(A ∪ B)))
50	47, 49	bitr3i 182	. . . 4 ⊢ (∀x ∈ A ∀y ∈ B suc (rank ‘⟨x, y⟩) ⊆ (rank ‘(A ∪ B)) ↔ (rank ‘(A × B)) ⊆ (rank ‘(A ∪ B)))
51	42, 50	sylib 205	. . 3 ⊢ (Lim (rank ‘(A ∪ B)) → (rank ‘(A × B)) ⊆ (rank ‘(A ∪ B)))
52	51	adantr 398	. 2 ⊢ ((Lim (rank ‘(A ∪ B)) ⋀ (A × B) ≠ ∅) → (rank ‘(A × B)) ⊆ (rank ‘(A ∪ B)))
53	3, 4	rankxpl 4772	. . 3 ⊢ ((A × B) ≠ ∅ → (rank ‘(A ∪ B)) ⊆ (rank ‘(A × B)))
54	53	adantl 397	. 2 ⊢ ((Lim (rank ‘(A ∪ B)) ⋀ (A × B) ≠ ∅) → (rank ‘(A ∪ B)) ⊆ (rank ‘(A × B)))
55	52, 54	eqssd 2130	1 ⊢ ((Lim (rank ‘(A ∪ B)) ⋀ (A × B) ≠ ∅) → (rank ‘(A × B)) = (rank ‘(A ∪ B)))

Syntax hints:   → wi 3   ↔ wb 153   ⋀ wa 230   = wceq 997   ∈ wcel 999   ≠ wne 1632  ∀wral 1692  Vcvv 1858   ∪ cun 2096   ⊆ wss 2098  ∅c0 2331  ℘cpw 2453  {cpr 2462  ⟨cop 2463  ∪cuni 2557  Oncon0 3005  Lim wlim 3006  suc csuc 3007   × cxp 3225   ‘cfv 3239  rankcrnk 4704

This theorem is referenced by:  rankxplim3 4776

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922  ax-reg 4653  ax-inf2 4687

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-rab 1699  df-v 1859  df-sbc 1989  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-if 2414  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-int 2588  df-iun 2622  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-id 2891  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-lim 3010  df-suc 3011  df-om 3189  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-fv 3255  df-rdg 3990  df-r1 4705  df-rank 4706

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