Theorem finsschain 9348

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 24364 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 3933	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑎 ⊆ ∪ 𝐴 ↔ ∅ ⊆ ∪ 𝐴))
2		sseq1 3933	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝑧 ↔ ∅ ⊆ 𝑧))
3	2	rexbidv 3186	. . . . . 6 ⊢ (𝑎 = ∅ → (∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧))
4	1, 3	imbi12d 347	. . . . 5 ⊢ (𝑎 = ∅ → ((𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧) ↔ (∅ ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧)))
5	4	imbi2d 343	. . . 4 ⊢ (𝑎 = ∅ → (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (∅ ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧))))
6		sseq1 3933	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ ∪ 𝐴 ↔ 𝑏 ⊆ ∪ 𝐴))
7		sseq1 3933	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ 𝑧 ↔ 𝑏 ⊆ 𝑧))
8	7	rexbidv 3186	. . . . . 6 ⊢ (𝑎 = 𝑏 → (∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧))
9	6, 8	imbi12d 347	. . . . 5 ⊢ (𝑎 = 𝑏 → ((𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧) ↔ (𝑏 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧)))
10	9	imbi2d 343	. . . 4 ⊢ (𝑎 = 𝑏 → (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝑏 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧))))
11		sseq1 3933	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ ∪ 𝐴 ↔ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴))
12		sseq1 3933	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ 𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝑧))
13	12	rexbidv 3186	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
14	11, 13	imbi12d 347	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧) ↔ ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
15	14	imbi2d 343	. . . 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 3933	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑎 ⊆ ∪ 𝐴 ↔ 𝐵 ⊆ ∪ 𝐴))
17		sseq1 3933	. . . . . . 7 ⊢ (𝑎 = 𝐵 → (𝑎 ⊆ 𝑧 ↔ 𝐵 ⊆ 𝑧))
18	17	rexbidv 3186	. . . . . 6 ⊢ (𝑎 = 𝐵 → (∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧))
19	16, 18	imbi12d 347	. . . . 5 ⊢ (𝑎 = 𝐵 → ((𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧) ↔ (𝐵 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧)))
20	19	imbi2d 343	. . . 4 ⊢ (𝑎 = 𝐵 → (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝑎 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑎 ⊆ 𝑧)) ↔ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝐵 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧))))
21		0ss 4349	. . . . . . . 8 ⊢ ∅ ⊆ 𝑧
22	21	rgenw 3080	. . . . . . 7 ⊢ ∀𝑧 ∈ 𝐴 ∅ ⊆ 𝑧
23		r19.2z 4454	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑧 ∈ 𝐴 ∅ ⊆ 𝑧) → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧)
24	22, 23	mpan2 704	. . . . . 6 ⊢ (𝐴 ≠ ∅ → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧)
25	24	adantr 486	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧)
26	25	a1d 26	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (∅ ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 ∅ ⊆ 𝑧))
27		id 23	. . . . . . . . 9 ⊢ ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴)
28	27	unssad 4191	. . . . . . . 8 ⊢ ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → 𝑏 ⊆ ∪ 𝐴)
29	28	imim1i 64	. . . . . . 7 ⊢ ((𝑏 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧) → ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧))
30		sseq2 3934	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝑏 ⊆ 𝑧 ↔ 𝑏 ⊆ 𝑤))
31	30	cbvrexvw 3241	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧 ↔ ∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤)
32		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴)
33	32	unssbd 4192	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → {𝑐} ⊆ ∪ 𝐴)
34		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑐 ∈ V
35	34	snss 4744	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ∪ 𝐴 ↔ {𝑐} ⊆ ∪ 𝐴)
36	33, 35	sylibr 237	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → 𝑐 ∈ ∪ 𝐴)
37		eluni2 4870	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ∪ 𝐴 ↔ ∃𝑢 ∈ 𝐴 𝑐 ∈ 𝑢)
38	36, 37	sylib 221	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → ∃𝑢 ∈ 𝐴 𝑐 ∈ 𝑢)
39		reeanv 3234	. . . . . . . . . . . 12 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤) ↔ (∃𝑢 ∈ 𝐴 𝑐 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤))
40		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → [⊊] Or 𝐴)
41		simprlr 792	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → 𝑤 ∈ 𝐴)
42		simprll 791	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → 𝑢 ∈ 𝐴)
43		sorpssun 7746	. . . . . . . . . . . . . . . 16 ⊢ (( [⊊] Or 𝐴 ∧ (𝑤 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴)) → (𝑤 ∪ 𝑢) ∈ 𝐴)
44	40, 41, 42, 43	syl12anc 850	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → (𝑤 ∪ 𝑢) ∈ 𝐴)
45		simprrr 794	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → 𝑏 ⊆ 𝑤)
46		simprrl 793	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → 𝑐 ∈ 𝑢)
47	46	snssd 4746	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → {𝑐} ⊆ 𝑢)
48		unss12 4186	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ⊆ 𝑤 ∧ {𝑐} ⊆ 𝑢) → (𝑏 ∪ {𝑐}) ⊆ (𝑤 ∪ 𝑢))
49	45, 47, 48	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → (𝑏 ∪ {𝑐}) ⊆ (𝑤 ∪ 𝑢))
50		sseq2 3934	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑤 ∪ 𝑢) → ((𝑏 ∪ {𝑐}) ⊆ 𝑧 ↔ (𝑏 ∪ {𝑐}) ⊆ (𝑤 ∪ 𝑢)))
51	50	rspcev 3576	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∪ 𝑢) ∈ 𝐴 ∧ (𝑏 ∪ {𝑐}) ⊆ (𝑤 ∪ 𝑢)) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
52	44, 49, 51	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤))) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)
53	52	expr 462	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
54	53	rexlimdvva 3219	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → (∃𝑢 ∈ 𝐴 ∃𝑤 ∈ 𝐴 (𝑐 ∈ 𝑢 ∧ 𝑏 ⊆ 𝑤) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
55	39, 54	biimtrrid 246	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → ((∃𝑢 ∈ 𝐴 𝑐 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤) → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
56	38, 55	mpand 708	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → (∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
57	31, 56	biimtrid 245	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴) → (∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧))
58	57	ex 418	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → (∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
59	58	a2d 30	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧) → ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
60	29, 59	syl5 35	. . . . . 6 ⊢ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → ((𝑏 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑏 ⊆ 𝑧) → ((𝑏 ∪ {𝑐}) ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 (𝑏 ∪ {𝑐}) ⊆ 𝑧)))
61	60	a2i 15	. . . . 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 9180	. . 3 ⊢ (𝐵 ∈ Fin → ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝐵 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧)))
64	63	com12 33	. 2 ⊢ ((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) → (𝐵 ∈ Fin → (𝐵 ⊆ ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧)))
65	64	imp32 424	1 ⊢ (((𝐴 ≠ ∅ ∧ [⊊] Or 𝐴) ∧ (𝐵 ∈ Fin ∧ 𝐵 ⊆ ∪ 𝐴)) → ∃𝑧 ∈ 𝐴 𝐵 ⊆ 𝑧)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  {csn 4583  ∪ cuni 4866   Or wor 5557   [⊊] crpss 7738  Fincfn 8973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-rpss 7739  df-om 7878  df-en 8974  df-fin 8977

This theorem is used by:  alexsubALTlem2  24367