Theorem cfeq0 9943

Description: Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)

Assertion

Ref	Expression
cfeq0	⊢ (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))

Proof of Theorem cfeq0

Dummy variables 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cfval 9934	. . . 4 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2	1	eqeq1d 2740	. . 3 ⊢ (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} = ∅))
3		vex 3426	. . . . . . . . 9 ⊢ 𝑣 ∈ V
4		eqeq1 2742	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑣 = (card‘𝑦)))
5	4	anbi1d 629	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
6	5	exbidv 1925	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
7	3, 6	elab 3602	. . . . . . . 8 ⊢ (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
8		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘(card‘𝑦)))
9		cardidm 9648	. . . . . . . . . . . 12 ⊢ (card‘(card‘𝑦)) = (card‘𝑦)
10	8, 9	eqtrdi 2795	. . . . . . . . . . 11 ⊢ (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘𝑦))
11		eqeq2 2750	. . . . . . . . . . 11 ⊢ (𝑣 = (card‘𝑦) → ((card‘𝑣) = 𝑣 ↔ (card‘𝑣) = (card‘𝑦)))
12	10, 11	mpbird 256	. . . . . . . . . 10 ⊢ (𝑣 = (card‘𝑦) → (card‘𝑣) = 𝑣)
13	12	adantr 480	. . . . . . . . 9 ⊢ ((𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (card‘𝑣) = 𝑣)
14	13	exlimiv 1934	. . . . . . . 8 ⊢ (∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (card‘𝑣) = 𝑣)
15	7, 14	sylbi 216	. . . . . . 7 ⊢ (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → (card‘𝑣) = 𝑣)
16		cardon 9633	. . . . . . 7 ⊢ (card‘𝑣) ∈ On
17	15, 16	eqeltrrdi 2848	. . . . . 6 ⊢ (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → 𝑣 ∈ On)
18	17	ssriv 3921	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ On
19		onint0 7618	. . . . 5 ⊢ ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ On → (∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} = ∅ ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}))
20	18, 19	ax-mp 5	. . . 4 ⊢ (∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} = ∅ ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
21		0ex 5226	. . . . . 6 ⊢ ∅ ∈ V
22		eqeq1 2742	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
23	22	anbi1d 629	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
24	23	exbidv 1925	. . . . . 6 ⊢ (𝑥 = ∅ → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
25	21, 24	elab 3602	. . . . 5 ⊢ (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
26		onss 7611	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
27		sstr 3925	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ On) → 𝑦 ⊆ On)
28	27	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ On)
29	26, 28	sylan 579	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ On)
30	29	3adant2 1129	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ On)
31	30	3adant3r 1179	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑦 ⊆ On)
32		simp2 1135	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → ∅ = (card‘𝑦))
33		simp3 1136	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
34		eqcom 2745	. . . . . . . . . . . 12 ⊢ (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
35		vex 3426	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
36		onssnum 9727	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
37	35, 36	mpan 686	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ On → 𝑦 ∈ dom card)
38		cardnueq0 9653	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
39	37, 38	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
40	34, 39	syl5bb 282	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
41	40	biimpa 476	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
42		sseq1 3942	. . . . . . . . . . . 12 ⊢ (𝑦 = ∅ → (𝑦 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
43		rexeq 3334	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
44	43	ralbidv 3120	. . . . . . . . . . . 12 ⊢ (𝑦 = ∅ → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
45	42, 44	anbi12d 630	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (∅ ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)))
46	45	biimpa 476	. . . . . . . . . 10 ⊢ ((𝑦 = ∅ ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
47	41, 46	sylan 579	. . . . . . . . 9 ⊢ (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
48		rex0 4288	. . . . . . . . . . . . 13 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
49	48	rgenw 3075	. . . . . . . . . . . 12 ⊢ ∀𝑧 ∈ 𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
50		r19.2z 4422	. . . . . . . . . . . 12 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑧 ∈ 𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤) → ∃𝑧 ∈ 𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
51	49, 50	mpan2 687	. . . . . . . . . . 11 ⊢ (𝐴 ≠ ∅ → ∃𝑧 ∈ 𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
52		rexnal 3165	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ 𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 ↔ ¬ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
53	51, 52	sylib 217	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ → ¬ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
54	53	necon4ai 2974	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 → 𝐴 = ∅)
55	47, 54	simpl2im 503	. . . . . . . 8 ⊢ (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝐴 = ∅)
56	31, 32, 33, 55	syl21anc 834	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝐴 = ∅)
57	56	3expib 1120	. . . . . 6 ⊢ (𝐴 ∈ On → ((∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝐴 = ∅))
58	57	exlimdv 1937	. . . . 5 ⊢ (𝐴 ∈ On → (∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝐴 = ∅))
59	25, 58	syl5bi 241	. . . 4 ⊢ (𝐴 ∈ On → (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → 𝐴 = ∅))
60	20, 59	syl5bi 241	. . 3 ⊢ (𝐴 ∈ On → (∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} = ∅ → 𝐴 = ∅))
61	2, 60	sylbid 239	. 2 ⊢ (𝐴 ∈ On → ((cf‘𝐴) = ∅ → 𝐴 = ∅))
62		fveq2 6756	. . 3 ⊢ (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
63		cf0 9938	. . 3 ⊢ (cf‘∅) = ∅
64	62, 63	eqtrdi 2795	. 2 ⊢ (𝐴 = ∅ → (cf‘𝐴) = ∅)
65	61, 64	impbid1 224	1 ⊢ (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  ∩ cint 4876  dom cdm 5580  Oncon0 6251  ‘cfv 6418  cardccrd 9624  cfccf 9626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-er 8456  df-en 8692  df-dom 8693  df-card 9628  df-cf 9630

This theorem is referenced by: (None)