Theorem heiborlem1 38936

Description: Lemma for heibor 38946. We work with a fixed open cover 𝑈 throughout. The set 𝐾 is the set of all subsets of 𝑋 that admit no finite subcover of 𝑈. (We wish to prove that 𝐾 is empty.) If a set 𝐶 has no finite subcover, then any finite cover of 𝐶 must contain a set that also has no finite subcover. (Contributed by Jeff Madsen, 23-Jan-2014.)

Hypotheses

Ref	Expression
heibor.1	⊢ 𝐽 = (MetOpen‘𝐷)
heibor.3	⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
heiborlem1.4	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
heiborlem1	⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐾) → ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑢,𝑣,𝐷   𝑢,𝐵,𝑣   𝑢,𝐽,𝑣,𝑥   𝑢,𝑈,𝑣,𝑥   𝑢,𝐶,𝑣   𝑥,𝐾

Allowed substitution hints:   𝐴(𝑣, 𝑢)   𝐵(𝑥)   𝐶(𝑥)   𝐾(𝑣, 𝑢)

Proof of Theorem heiborlem1

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		heiborlem1.4	. . . . . . . 8 ⊢ 𝐵 ∈ V
2		sseq1 3938	. . . . . . . . . 10 ⊢ (𝑢 = 𝐵 → (𝑢 ⊆ ∪ 𝑣 ↔ 𝐵 ⊆ ∪ 𝑣))
3	2	rexbidv 3190	. . . . . . . . 9 ⊢ (𝑢 = 𝐵 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣))
4	3	notbid 321	. . . . . . . 8 ⊢ (𝑢 = 𝐵 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣))
5		heibor.3	. . . . . . . 8 ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
6	1, 4, 5	elab2 3639	. . . . . . 7 ⊢ (𝐵 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣)
7	6	con2bii 360	. . . . . 6 ⊢ (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣 ↔ ¬ 𝐵 ∈ 𝐾)
8	7	ralbii 3112	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝐵 ∈ 𝐾)
9		ralnex 3092	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝐵 ∈ 𝐾 ↔ ¬ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾)
10	8, 9	bitr2i 279	. . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾 ↔ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣)
11		unieq 4881	. . . . . . . . 9 ⊢ (𝑣 = (𝑡‘𝑥) → ∪ 𝑣 = ∪ (𝑡‘𝑥))
12	11	sseq2d 3945	. . . . . . . 8 ⊢ (𝑣 = (𝑡‘𝑥) → (𝐵 ⊆ ∪ 𝑣 ↔ 𝐵 ⊆ ∪ (𝑡‘𝑥)))
13	12	ac6sfi 9300	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣) → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥)))
14	13	ex 418	. . . . . 6 ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣 → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))))
15	14	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣 → ∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))))
16		sseq1 3938	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐶 → (𝑢 ⊆ ∪ 𝑣 ↔ 𝐶 ⊆ ∪ 𝑣))
17	16	rexbidv 3190	. . . . . . . . . . 11 ⊢ (𝑢 = 𝐶 → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣))
18	17	notbid 321	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣))
19	18, 5	elab2g 3637	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐾 → (𝐶 ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣))
20	19	ibi 270	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐾 → ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣)
21		frn 6736	. . . . . . . . . . . . . . 15 ⊢ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
22	21	ad2antrl 741	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ran 𝑡 ⊆ (𝒫 𝑈 ∩ Fin))
23		inss1 4136	. . . . . . . . . . . . . 14 ⊢ (𝒫 𝑈 ∩ Fin) ⊆ 𝒫 𝑈
24	22, 23	sstrdi 3925	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ran 𝑡 ⊆ 𝒫 𝑈)
25		sspwuni 5063	. . . . . . . . . . . . 13 ⊢ (ran 𝑡 ⊆ 𝒫 𝑈 ↔ ∪ ran 𝑡 ⊆ 𝑈)
26	24, 25	sylib 221	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ⊆ 𝑈)
27		vex 3458	. . . . . . . . . . . . . . 15 ⊢ 𝑡 ∈ V
28	27	rnex 7943	. . . . . . . . . . . . . 14 ⊢ ran 𝑡 ∈ V
29	28	uniex 7779	. . . . . . . . . . . . 13 ⊢ ∪ ran 𝑡 ∈ V
30	29	elpw 4564	. . . . . . . . . . . 12 ⊢ (∪ ran 𝑡 ∈ 𝒫 𝑈 ↔ ∪ ran 𝑡 ⊆ 𝑈)
31	26, 30	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ∈ 𝒫 𝑈)
32		ffn 6728	. . . . . . . . . . . . . . 15 ⊢ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) → 𝑡 Fn 𝐴)
33	32	ad2antrl 741	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → 𝑡 Fn 𝐴)
34		dffn4 6821	. . . . . . . . . . . . . 14 ⊢ (𝑡 Fn 𝐴 ↔ 𝑡:𝐴⟶ontoran 𝑡)
35	33, 34	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → 𝑡:𝐴⟶ontoran 𝑡)
36		fofi 9330	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶ontoran 𝑡) → ran 𝑡 ∈ Fin)
37	35, 36	syldan 603	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ran 𝑡 ∈ Fin)
38		inss2 4137	. . . . . . . . . . . . 13 ⊢ (𝒫 𝑈 ∩ Fin) ⊆ Fin
39	22, 38	sstrdi 3925	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ran 𝑡 ⊆ Fin)
40		unifi 9358	. . . . . . . . . . . 12 ⊢ ((ran 𝑡 ∈ Fin ∧ ran 𝑡 ⊆ Fin) → ∪ ran 𝑡 ∈ Fin)
41	37, 39, 40	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ∈ Fin)
42	31, 41	elind 4100	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
43	42	adantlr 728	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin))
44		simplr 781	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
45		fnfvelrn 7101	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑡 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑡‘𝑥) ∈ ran 𝑡)
46	32, 45	sylan 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ 𝑥 ∈ 𝐴) → (𝑡‘𝑥) ∈ ran 𝑡)
47	46	adantll 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥 ∈ 𝐴) → (𝑡‘𝑥) ∈ ran 𝑡)
48		elssuni 4902	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡‘𝑥) ∈ ran 𝑡 → (𝑡‘𝑥) ⊆ ∪ ran 𝑡)
49		uniss 4878	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡‘𝑥) ⊆ ∪ ran 𝑡 → ∪ (𝑡‘𝑥) ⊆ ∪ ∪ ran 𝑡)
50	47, 48, 49	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥 ∈ 𝐴) → ∪ (𝑡‘𝑥) ⊆ ∪ ∪ ran 𝑡)
51		sstr2 3920	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ⊆ ∪ (𝑡‘𝑥) → (∪ (𝑡‘𝑥) ⊆ ∪ ∪ ran 𝑡 → 𝐵 ⊆ ∪ ∪ ran 𝑡))
52	50, 51	syl5com 32	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) ∧ 𝑥 ∈ 𝐴) → (𝐵 ⊆ ∪ (𝑡‘𝑥) → 𝐵 ⊆ ∪ ∪ ran 𝑡))
53	52	ralimdva 3178	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin)) → (∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥) → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡))
54	53	impr 460	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡)
55		iunss 5006	. . . . . . . . . . . 12 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡)
56	54, 55	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡)
57	56	adantlr 728	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ ∪ ran 𝑡)
58	44, 57	sstrd 3923	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → 𝐶 ⊆ ∪ ∪ ran 𝑡)
59		unieq 4881	. . . . . . . . . . 11 ⊢ (𝑣 = ∪ ran 𝑡 → ∪ 𝑣 = ∪ ∪ ran 𝑡)
60	59	sseq2d 3945	. . . . . . . . . 10 ⊢ (𝑣 = ∪ ran 𝑡 → (𝐶 ⊆ ∪ 𝑣 ↔ 𝐶 ⊆ ∪ ∪ ran 𝑡))
61	60	rspcev 3580	. . . . . . . . 9 ⊢ ((∪ ran 𝑡 ∈ (𝒫 𝑈 ∩ Fin) ∧ 𝐶 ⊆ ∪ ∪ ran 𝑡) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣)
62	43, 58, 61	syl2anc 596	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐶 ⊆ ∪ 𝑣)
63	20, 62	nsyl3 139	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ (𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥))) → ¬ 𝐶 ∈ 𝐾)
64	63	ex 418	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → ((𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥)) → ¬ 𝐶 ∈ 𝐾))
65	64	exlimdv 1968	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (∃𝑡(𝑡:𝐴⟶(𝒫 𝑈 ∩ Fin) ∧ ∀𝑥 ∈ 𝐴 𝐵 ⊆ ∪ (𝑡‘𝑥)) → ¬ 𝐶 ∈ 𝐾))
66	15, 65	syld 48	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (∀𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝐵 ⊆ ∪ 𝑣 → ¬ 𝐶 ∈ 𝐾))
67	10, 66	biimtrid 245	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (¬ ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾 → ¬ 𝐶 ∈ 𝐾))
68	67	con4d 116	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) → (𝐶 ∈ 𝐾 → ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾))
69	68	3impia 1135	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐾) → ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ∪ cuni 4870  ∪ ciun 4954  ran crn 5656   Fn wfn 6550  ⟶wf 6551  ⟶ontowfo 6553  ‘cfv 6555  Fincfn 8998  MetOpencmopn 21695

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-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

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-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-iun 4956  df-br 5107  df-opab 5171  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-om 7899  df-1o 8501  df-en 8999  df-dom 9000  df-fin 9002

This theorem is used by:  heiborlem3  38938  heiborlem10  38945