Theorem finsschain 9399

Description: A finite subset of the union of a superset chain is a subset of some element of the chain. A useful preliminary result for alexsub 24053 and others. (Contributed by Jeff Hankins, 25-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 18-May-2015.)

Assertion

Ref	Expression
finsschain	⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝐵 ∈ Fin ∧ 𝐵 ⊆ ∪ 𝐴)) → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧)

Distinct variable groups:   𝑧,𝐴   𝑧,𝐵

Proof of Theorem finsschain

Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 4009	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑎 ⊆ ∪ 𝐴 ↔ ∅ ⊆ ∪ 𝐴))
2		sseq1 4009	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝑧 ↔ ∅ ⊆ 𝑧))
3	2	rexbidv 3179	. . . . . 6 ⊢ (𝑎 = ∅ → (∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧))
4	1, 3	imbi12d 344	. . . . 5 ⊢ (𝑎 = ∅ → ((𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧) ↔ (∅ ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧)))
5	4	imbi2d 340	. . . 4 ⊢ (𝑎 = ∅ → (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (∅ ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧))))
6		sseq1 4009	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ ∪ 𝐴 ↔ 𝑏 ⊆ ∪ 𝐴))
7		sseq1 4009	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ 𝑧 ↔ 𝑏 ⊆ 𝑧))
8	7	rexbidv 3179	. . . . . 6 ⊢ (𝑎 = 𝑏 → (∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧))
9	6, 8	imbi12d 344	. . . . 5 ⊢ (𝑎 = 𝑏 → ((𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧) ↔ (𝑏 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧)))
10	9	imbi2d 340	. . . 4 ⊢ (𝑎 = 𝑏 → (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝑏 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧))))
11		sseq1 4009	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ ∪ 𝐴 ↔ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴))
12		sseq1 4009	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ 𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝑧))
13	12	rexbidv 3179	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
14	11, 13	imbi12d 344	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧) ↔ ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
15	14	imbi2d 340	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))))
16		sseq1 4009	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑎 ⊆ ∪ 𝐴 ↔ 𝐵 ⊆ ∪ 𝐴))
17		sseq1 4009	. . . . . . 7 ⊢ (𝑎 = 𝐵 → (𝑎 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑧))
18	17	rexbidv 3179	. . . . . 6 ⊢ (𝑎 = 𝐵 → (∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧))
19	16, 18	imbi12d 344	. . . . 5 ⊢ (𝑎 = 𝐵 → ((𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧) ↔ (𝐵 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧)))
20	19	imbi2d 340	. . . 4 ⊢ (𝑎 = 𝐵 → (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝐵 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧))))
21		0ss 4400	. . . . . . . 8 ⊢ ∅ ⊆ 𝑧
22	21	rgenw 3065	. . . . . . 7 ⊢ ∀𝑧 ∈ 𝐴 ∅ ⊆ 𝑧
23		r19.2z 4495	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑧 ∈ 𝐴 ∅ ⊆ 𝑧) → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧)
24	22, 23	mpan2 691	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧)
25	24	adantr 480	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧)
26	25	a1d 25	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (∅ ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧))
27		id 22	. . . . . . . . 9 ⊢ ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴)
28	27	unssad 4193	. . . . . . . 8 ⊢ ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → 𝑏 ⊆ ∪ 𝐴)
29	28	imim1i 63	. . . . . . 7 ⊢ ((𝑏 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧) → ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧))
30		sseq2 4010	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝑏 ⊆ 𝑧 ↔ 𝑏 ⊆ 𝑤))
31	30	cbvrexvw 3238	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧 ↔ ∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤)
32		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴)
33	32	unssbd 4194	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → {𝑐} ⊆ ∪ 𝐴)
34		vex 3484	. . . . . . . . . . . . . 14 ⊢ 𝑐 ∈ V
35	34	snss 4785	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ∪ 𝐴 ↔ {𝑐} ⊆ ∪ 𝐴)
36	33, 35	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → 𝑐 ∈ ∪ 𝐴)
37		eluni2 4911	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ∪ 𝐴 ↔ ∃𝑢 ∈ 𝐴 𝑐 ∈ 𝑢)
38	36, 37	sylib 218	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → ∃𝑢 ∈ 𝐴 𝑐 ∈ 𝑢)
39		reeanv 3229	. . . . . . . . . . . 12 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤) ↔ (∃𝑢 ∈ 𝐴 𝑐 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤))
40		simpllr 776	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → [⊊] Or 𝐴)
41		simprlr 780	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → 𝑤 ∈ 𝐴)
42		simprll 779	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → 𝑢 ∈ 𝐴)
43		sorpssun 7750	. . . . . . . . . . . . . . . 16 ⊢ (( [⊊] Or 𝐴 ∧ (𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → (𝑤 ∪ 𝑢) ∈ 𝐴)
44	40, 41, 42, 43	syl12anc 837	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → (𝑤 ∪ 𝑢) ∈ 𝐴)
45		simprrr 782	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → 𝑏 ⊆ 𝑤)
46		simprrl 781	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → 𝑐 ∈ 𝑢)
47	46	snssd 4809	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → {𝑐} ⊆ 𝑢)
48		unss12 4188	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ⊆ 𝑤 ∧ {𝑐} ⊆ 𝑢) → (𝑏 ∪ {𝑐}) ⊆ (𝑤 ∪ 𝑢))
49	45, 47, 48	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → (𝑏 ∪ {𝑐}) ⊆ (𝑤 ∪ 𝑢))
50		sseq2 4010	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∪ 𝑢) → ((𝑏 ∪ {𝑐}) ⊆ 𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑤 ∪ 𝑢)))
51	50	rspcev 3622	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∪ 𝑢) ∈ 𝐴 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑤 ∪ 𝑢)) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
52	44, 49, 51	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
53	52	expr 456	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
54	53	rexlimdvva 3213	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → (∃𝑢 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
55	39, 54	biimtrrid 243	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → ((∃𝑢 ∈ 𝐴 𝑐 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
56	38, 55	mpand 695	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → (∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
57	31, 56	biimtrid 242	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → (∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
58	57	ex 412	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → (∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
59	58	a2d 29	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧) → ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
60	29, 59	syl5 34	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → ((𝑏 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧) → ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
61	60	a2i 14	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝑏 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧)) → ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
62	61	a1i 11	. . . 4 ⊢ (𝑏 ∈ Fin → (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝑏 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧)) → ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))))
63	5, 10, 15, 20, 26, 62	findcard2 9204	. . 3 ⊢ (𝐵 ∈ Fin → ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝐵 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧)))
64	63	com12 32	. 2 ⊢ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝐵 ∈ Fin → (𝐵 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧)))
65	64	imp32 418	1 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝐵 ∈ Fin ∧ 𝐵 ⊆ ∪ 𝐴)) → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∪ cun 3949   ⊆ wss 3951  ∅c0 4333  {csn 4626  ∪ cuni 4907   Or wor 5591   [⊊] crpss 7742  Fincfn 8985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-rpss 7743  df-om 7888  df-en 8986  df-fin 8989

This theorem is referenced by:  alexsubALTlem2  24056