Theorem finsschain 9056

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 23104 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 3942	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑎 ⊆ ∪ 𝐴 ↔ ∅ ⊆ ∪ 𝐴))
2		sseq1 3942	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝑧 ↔ ∅ ⊆ 𝑧))
3	2	rexbidv 3225	. . . . . 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 3942	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ ∪ 𝐴 ↔ 𝑏 ⊆ ∪ 𝐴))
7		sseq1 3942	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ 𝑧 ↔ 𝑏 ⊆ 𝑧))
8	7	rexbidv 3225	. . . . . 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 3942	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ ∪ 𝐴 ↔ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴))
12		sseq1 3942	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ 𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝑧))
13	12	rexbidv 3225	. . . . . 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 3942	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑎 ⊆ ∪ 𝐴 ↔ 𝐵 ⊆ ∪ 𝐴))
17		sseq1 3942	. . . . . . 7 ⊢ (𝑎 = 𝐵 → (𝑎 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑧))
18	17	rexbidv 3225	. . . . . 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 4327	. . . . . . . 8 ⊢ ∅ ⊆ 𝑧
22	21	rgenw 3075	. . . . . . 7 ⊢ ∀𝑧 ∈ 𝐴 ∅ ⊆ 𝑧
23		r19.2z 4422	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑧 ∈ 𝐴 ∅ ⊆ 𝑧) → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧)
24	22, 23	mpan2 687	. . . . . 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 4117	. . . . . . . 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 3943	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝑏 ⊆ 𝑧 ↔ 𝑏 ⊆ 𝑤))
31	30	cbvrexvw 3373	. . . . . . . . . 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 4118	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → {𝑐} ⊆ ∪ 𝐴)
34		vex 3426	. . . . . . . . . . . . . 14 ⊢ 𝑐 ∈ V
35	34	snss 4716	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ∪ 𝐴 ↔ {𝑐} ⊆ ∪ 𝐴)
36	33, 35	sylibr 233	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → 𝑐 ∈ ∪ 𝐴)
37		eluni2 4840	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ∪ 𝐴 ↔ ∃𝑢 ∈ 𝐴 𝑐 ∈ 𝑢)
38	36, 37	sylib 217	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → ∃𝑢 ∈ 𝐴 𝑐 ∈ 𝑢)
39		reeanv 3292	. . . . . . . . . . . 12 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤) ↔ (∃𝑢 ∈ 𝐴 𝑐 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤))
40		simpllr 772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → [⊊] Or 𝐴)
41		simprlr 776	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → 𝑤 ∈ 𝐴)
42		simprll 775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → 𝑢 ∈ 𝐴)
43		sorpssun 7561	. . . . . . . . . . . . . . . 16 ⊢ (( [⊊] Or 𝐴 ∧ (𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → (𝑤 ∪ 𝑢) ∈ 𝐴)
44	40, 41, 42, 43	syl12anc 833	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → (𝑤 ∪ 𝑢) ∈ 𝐴)
45		simprrr 778	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → 𝑏 ⊆ 𝑤)
46		simprrl 777	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → 𝑐 ∈ 𝑢)
47	46	snssd 4739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → {𝑐} ⊆ 𝑢)
48		unss12 4112	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ⊆ 𝑤 ∧ {𝑐} ⊆ 𝑢) → (𝑏 ∪ {𝑐}) ⊆ (𝑤 ∪ 𝑢))
49	45, 47, 48	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → (𝑏 ∪ {𝑐}) ⊆ (𝑤 ∪ 𝑢))
50		sseq2 3943	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∪ 𝑢) → ((𝑏 ∪ {𝑐}) ⊆ 𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑤 ∪ 𝑢)))
51	50	rspcev 3552	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∪ 𝑢) ∈ 𝐴 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑤 ∪ 𝑢)) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
52	44, 49, 51	syl2anc 583	. . . . . . . . . . . . . 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 3222	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → (∃𝑢 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
55	39, 54	syl5bir 242	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → ((∃𝑢 ∈ 𝐴 𝑐 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
56	38, 55	mpand 691	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → (∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
57	31, 56	syl5bi 241	. . . . . . . . 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 8909	. . 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 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∪ cuni 4836   Or wor 5493   [⊊] crpss 7553  Fincfn 8691

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-sep 5218  ax-nul 5225  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-rab 3072  df-v 3424  df-sbc 3712  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-br 5071  df-opab 5133  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  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-ord 6254  df-on 6255  df-lim 6256  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-rpss 7554  df-om 7688  df-en 8692  df-fin 8695

This theorem is referenced by:  alexsubALTlem2  23107