Theorem cfeq0 10364

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 10354	. . . 4 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2	1	eqeq1d 2765	. . 3 ⊢ (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} = ∅))
3		vex 3458	. . . . . . . . 9 ⊢ 𝑣 ∈ V
4		eqeq1 2767	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑣 = (card‘𝑦)))
5	4	anbi1d 643	. . . . . . . . . 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 1956	. . . . . . . . 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 3636	. . . . . . . 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 6911	. . . . . . . . . . . 12 ⊢ (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘(card‘𝑦)))
9		cardidm 10070	. . . . . . . . . . . 12 ⊢ (card‘(card‘𝑦)) = (card‘𝑦)
10	8, 9	eqtrdi 2814	. . . . . . . . . . 11 ⊢ (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘𝑦))
11		eqeq2 2775	. . . . . . . . . . 11 ⊢ (𝑣 = (card‘𝑦) → ((card‘𝑣) = 𝑣 ↔ (card‘𝑣) = (card‘𝑦)))
12	10, 11	mpbird 260	. . . . . . . . . 10 ⊢ (𝑣 = (card‘𝑦) → (card‘𝑣) = 𝑣)
13	12	adantr 486	. . . . . . . . 9 ⊢ ((𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (card‘𝑣) = 𝑣)
14	13	exlimiv 1965	. . . . . . . 8 ⊢ (∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (card‘𝑣) = 𝑣)
15	7, 14	sylbi 220	. . . . . . 7 ⊢ (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → (card‘𝑣) = 𝑣)
16		cardon 10055	. . . . . . 7 ⊢ (card‘𝑣) ∈ On
17	15, 16	eqeltrrdi 2873	. . . . . 6 ⊢ (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → 𝑣 ∈ On)
18	17	ssriv 3917	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ On
19		onint0 7832	. . . . 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 5265	. . . . . 6 ⊢ ∅ ∈ V
22		eqeq1 2767	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
23	22	anbi1d 643	. . . . . . 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 1956	. . . . . 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 3636	. . . . 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 7826	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
27		sstr 3921	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ On) → 𝑦 ⊆ On)
28	27	ancoms 464	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ On)
29	26, 28	sylan 592	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ On)
30	29	3adant2 1149	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ On)
31	30	3adant3r 1200	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑦 ⊆ On)
32		simp2 1155	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → ∅ = (card‘𝑦))
33		simp3 1156	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
34		eqcom 2770	. . . . . . . . . . . 12 ⊢ (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
35		vex 3458	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
36		onssnum 10149	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
37	35, 36	mpan 703	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ On → 𝑦 ∈ dom card)
38		cardnueq0 10075	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
39	37, 38	syl 18	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
40	34, 39	bitrid 286	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
41	40	biimpa 482	. . . . . . . . . 10 ⊢ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
42		sseq1 3938	. . . . . . . . . . . 12 ⊢ (𝑦 = ∅ → (𝑦 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
43		rexeq 3319	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
44	43	ralbidv 3189	. . . . . . . . . . . 12 ⊢ (𝑦 = ∅ → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
45	42, 44	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (∅ ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)))
46	45	biimpa 482	. . . . . . . . . 10 ⊢ ((𝑦 = ∅ ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
47	41, 46	sylan 592	. . . . . . . . 9 ⊢ (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (∅ ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
48		rex0 4311	. . . . . . . . . . . . 13 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
49	48	rgenw 3084	. . . . . . . . . . . 12 ⊢ ∀𝑧 ∈ 𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
50		r19.2z 4458	. . . . . . . . . . . 12 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑧 ∈ 𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤) → ∃𝑧 ∈ 𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
51	49, 50	mpan2 704	. . . . . . . . . . 11 ⊢ (𝐴 ≠ ∅ → ∃𝑧 ∈ 𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
52		rexnal 3118	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ 𝐴 ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 ↔ ¬ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
53	51, 52	sylib 221	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ → ¬ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
54	53	necon4ai 2990	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 → 𝐴 = ∅)
55	47, 54	simpl2im 513	. . . . . . . 8 ⊢ (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝐴 = ∅)
56	31, 32, 33, 55	syl21anc 851	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝐴 = ∅)
57	56	3expib 1140	. . . . . 6 ⊢ (𝐴 ∈ On → ((∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝐴 = ∅))
58	57	exlimdv 1968	. . . . 5 ⊢ (𝐴 ∈ On → (∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝐴 = ∅))
59	25, 58	biimtrid 245	. . . 4 ⊢ (𝐴 ∈ On → (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → 𝐴 = ∅))
60	20, 59	biimtrid 245	. . 3 ⊢ (𝐴 ∈ On → (∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} = ∅ → 𝐴 = ∅))
61	2, 60	sylbid 243	. 2 ⊢ (𝐴 ∈ On → ((cf‘𝐴) = ∅ → 𝐴 = ∅))
62		fveq2 6911	. . 3 ⊢ (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
63		cf0 10358	. . 3 ⊢ (cf‘∅) = ∅
64	62, 63	eqtrdi 2814	. 2 ⊢ (𝐴 = ∅ → (cf‘𝐴) = ∅)
65	61, 64	impbid1 228	1 ⊢ (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ∩ cint 4910  dom cdm 5656  Oncon0 6383  ‘cfv 6561  cardccrd 10046  cfccf 10048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7403  df-ov 7449  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-er 8748  df-en 9005  df-dom 9006  df-card 10050  df-cf 10052

This theorem is used by: (None)