atcvat4i - Hilbert Space Explorer

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Theorem atcvat4i 32429

Description: A condition implying existence of an atom with the properties shown. Lemma 3.2.20 of [PtakPulmannova] p. 68. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
atcvat3.1	⊢ 𝐴 ∈ C_ℋ

**Assertion**
Ref	Expression
atcvat4i	⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶

**Proof of Theorem atcvat4i**
Step	Hyp	Ref	Expression
1		atcvat3.1	. . . . . . . . 9 ⊢ 𝐴 ∈ C_ℋ
2	1	hatomici 32391	. . . . . . . 8 ⊢ (𝐴 ≠ 0_ℋ → ∃𝑥 ∈ HAtoms 𝑥 ⊆ 𝐴)
3		atelch 32376	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ HAtoms → 𝐶 ∈ C_ℋ )
4		atelch 32376	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ C_ℋ )
5		chub1 31539	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → 𝐶 ⊆ (𝐶 ∨_ℋ 𝑥))
6	3, 4, 5	syl2an 595	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ HAtoms ∧ 𝑥 ∈ HAtoms) → 𝐶 ⊆ (𝐶 ∨_ℋ 𝑥))
7		sseq1 4034	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝐶 → (𝐵 ⊆ (𝐶 ∨_ℋ 𝑥) ↔ 𝐶 ⊆ (𝐶 ∨_ℋ 𝑥)))
8	6, 7	imbitrrid 246	. . . . . . . . . . . . 13 ⊢ (𝐵 = 𝐶 → ((𝐶 ∈ HAtoms ∧ 𝑥 ∈ HAtoms) → 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
9	8	expd 415	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐶 → (𝐶 ∈ HAtoms → (𝑥 ∈ HAtoms → 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
10	9	impcom 407	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → (𝑥 ∈ HAtoms → 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
11	10	anim2d 611	. . . . . . . . . 10 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ∈ HAtoms) → (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
12	11	expcomd 416	. . . . . . . . 9 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → (𝑥 ∈ HAtoms → (𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
13	12	reximdvai 3171	. . . . . . . 8 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → (∃𝑥 ∈ HAtoms 𝑥 ⊆ 𝐴 → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
14	2, 13	syl5 34	. . . . . . 7 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → (𝐴 ≠ 0_ℋ → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
15	14	ex 412	. . . . . 6 ⊢ (𝐶 ∈ HAtoms → (𝐵 = 𝐶 → (𝐴 ≠ 0_ℋ → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
16	15	a1i 11	. . . . 5 ⊢ (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) → (𝐶 ∈ HAtoms → (𝐵 = 𝐶 → (𝐴 ≠ 0_ℋ → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))))
17	16	com4l 92	. . . 4 ⊢ (𝐶 ∈ HAtoms → (𝐵 = 𝐶 → (𝐴 ≠ 0_ℋ → (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))))
18	17	imp4a 422	. . 3 ⊢ (𝐶 ∈ HAtoms → (𝐵 = 𝐶 → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
19	18	adantl 481	. 2 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐵 = 𝐶 → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
20		atelch 32376	. . . . . . . 8 ⊢ (𝐵 ∈ HAtoms → 𝐵 ∈ C_ℋ )
21		chlejb2 31545	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → (𝐶 ⊆ 𝐴 ↔ (𝐴 ∨_ℋ 𝐶) = 𝐴))
22	1, 21	mpan2 690	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ C_ℋ → (𝐶 ⊆ 𝐴 ↔ (𝐴 ∨_ℋ 𝐶) = 𝐴))
23	22	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐶 ⊆ 𝐴) → (𝐴 ∨_ℋ 𝐶) = 𝐴)
24	23	sseq2d 4041	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐶 ⊆ 𝐴) → (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) ↔ 𝐵 ⊆ 𝐴))
25	24	biimpa 476	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ C_ℋ ∧ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ 𝐴)
26	25	expl 457	. . . . . . . . . 10 ⊢ (𝐶 ∈ C_ℋ → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ 𝐴))
27	26	adantl 481	. . . . . . . . 9 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ 𝐴))
28		chub2 31540	. . . . . . . . 9 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))
29	27, 28	jctird 526	. . . . . . . 8 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
30	20, 3, 29	syl2an 595	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
31		simpl 482	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → 𝐵 ∈ HAtoms)
32	30, 31	jctild 525	. . . . . 6 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ∈ HAtoms ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵)))))
33	32	impl 455	. . . . 5 ⊢ ((((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ∈ HAtoms ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
34		sseq1 4034	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
35		oveq2 7456	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐶 ∨_ℋ 𝑥) = (𝐶 ∨_ℋ 𝐵))
36	35	sseq2d 4041	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐵 ⊆ (𝐶 ∨_ℋ 𝑥) ↔ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵)))
37	34, 36	anbi12d 631	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
38	37	rspcev 3635	. . . . 5 ⊢ ((𝐵 ∈ HAtoms ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
39	33, 38	syl 17	. . . 4 ⊢ ((((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
40	39	adantrl 715	. . 3 ⊢ ((((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ 𝐶 ⊆ 𝐴) ∧ (𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
41	40	exp31 419	. 2 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐶 ⊆ 𝐴 → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
42		simpr 484	. . 3 ⊢ ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))
43		ioran 984	. . . 4 ⊢ (¬ (𝐵 = 𝐶 ∨ 𝐶 ⊆ 𝐴) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴))
44	1	atcvat3i 32428	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms))
45	3	ad2antlr 726	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → 𝐶 ∈ C_ℋ )
46	44	imp 406	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms)
47		simpll 766	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → 𝐵 ∈ HAtoms)
48	45, 46, 47	3jca 1128	. . . . . . . . . 10 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → (𝐶 ∈ C_ℋ ∧ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms ∧ 𝐵 ∈ HAtoms))
49		inss2 4259	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐵 ∨_ℋ 𝐶)
50		chjcom 31538	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝐶) = (𝐶 ∨_ℋ 𝐵))
51	20, 3, 50	syl2an 595	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐵 ∨_ℋ 𝐶) = (𝐶 ∨_ℋ 𝐵))
52	49, 51	sseqtrid 4061	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐶 ∨_ℋ 𝐵))
53	52	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐶 ∨_ℋ 𝐵))
54		atnssm0 32408	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐶 ∈ HAtoms) → (¬ 𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 0_ℋ))
55	1, 54	mpan 689	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ HAtoms → (¬ 𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 0_ℋ))
56	55	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (¬ 𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 0_ℋ))
57		inss1 4258	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴
58		sslin 4264	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴 → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ (𝐶 ∩ 𝐴))
59	57, 58	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ (𝐶 ∩ 𝐴)
60		incom 4230	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∩ 𝐴) = (𝐴 ∩ 𝐶)
61	59, 60	sseqtri 4045	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ (𝐴 ∩ 𝐶)
62		sseq2 4035	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∩ 𝐶) = 0_ℋ → ((𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ (𝐴 ∩ 𝐶) ↔ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ 0_ℋ))
63	61, 62	mpbii 233	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝐶) = 0_ℋ → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ 0_ℋ)
64		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → 𝐶 ∈ C_ℋ )
65		chjcl 31389	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝐶) ∈ C_ℋ )
66		chincl 31531	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ C_ℋ ∧ (𝐵 ∨_ℋ 𝐶) ∈ C_ℋ ) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ C_ℋ )
67	1, 65, 66	sylancr 586	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ C_ℋ )
68		chincl 31531	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ C_ℋ ∧ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ C_ℋ ) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ )
69	64, 67, 68	syl2anc 583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ )
70	20, 3, 69	syl2an 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ )
71		chle0 31475	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ → ((𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ 0_ℋ ↔ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ))
72	70, 71	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ 0_ℋ ↔ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ))
73	63, 72	imbitrid 244	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ∩ 𝐶) = 0_ℋ → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ))
74	56, 73	sylbid 240	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (¬ 𝐶 ⊆ 𝐴 → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ))
75	74	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ¬ 𝐶 ⊆ 𝐴) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ)
76	75	adantrl 715	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴)) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ)
77	76	adantrr 716	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ)
78	53, 77	jca 511	. . . . . . . . . 10 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐶 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ))
79		atexch 32413	. . . . . . . . . 10 ⊢ ((𝐶 ∈ C_ℋ ∧ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms ∧ 𝐵 ∈ HAtoms) → (((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐶 ∨_ℋ 𝐵) ∧ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ) → 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)))))
80	48, 78, 79	sylc 65	. . . . . . . . 9 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))
81	80, 57	jctil 519	. . . . . . . 8 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)))))
82	81	ex 412	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))))
83	44, 82	jcad 512	. . . . . 6 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms ∧ ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)))))))
84		sseq1 4034	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝑥 ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴))
85		oveq2 7456	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝐶 ∨_ℋ 𝑥) = (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))
86	85	sseq2d 4041	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝐵 ⊆ (𝐶 ∨_ℋ 𝑥) ↔ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)))))
87	84, 86	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → ((𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)) ↔ ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))))
88	87	rspcev 3635	. . . . . 6 ⊢ (((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms ∧ ((𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
89	83, 88	syl6 35	. . . . 5 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
90	89	expd 415	. . . 4 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) → (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
91	43, 90	biimtrid 242	. . 3 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (¬ (𝐵 = 𝐶 ∨ 𝐶 ⊆ 𝐴) → (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
92	42, 91	syl7 74	. 2 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (¬ (𝐵 = 𝐶 ∨ 𝐶 ⊆ 𝐴) → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
93	19, 41, 92	ecase3d 1034	1 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∃wrex 3076 ∩ cin 3975 ⊆ wss 3976 (class class class)co 7448 C_ℋ cch 30961 ∨_ℋ chj 30965 0_ℋc0h 30967 HAtomscat 30997

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264 ax-hilex 31031 ax-hfvadd 31032 ax-hvcom 31033 ax-hvass 31034 ax-hv0cl 31035 ax-hvaddid 31036 ax-hfvmul 31037 ax-hvmulid 31038 ax-hvmulass 31039 ax-hvdistr1 31040 ax-hvdistr2 31041 ax-hvmul0 31042 ax-hfi 31111 ax-his1 31114 ax-his2 31115 ax-his3 31116 ax-his4 31117 ax-hcompl 31234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-cn 23256 df-cnp 23257 df-lm 23258 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cfil 25308 df-cau 25309 df-cmet 25310 df-grpo 30525 df-gid 30526 df-ginv 30527 df-gdiv 30528 df-ablo 30577 df-vc 30591 df-nv 30624 df-va 30627 df-ba 30628 df-sm 30629 df-0v 30630 df-vs 30631 df-nmcv 30632 df-ims 30633 df-dip 30733 df-ssp 30754 df-ph 30845 df-cbn 30895 df-hnorm 31000 df-hba 31001 df-hvsub 31003 df-hlim 31004 df-hcau 31005 df-sh 31239 df-ch 31253 df-oc 31284 df-ch0 31285 df-shs 31340 df-span 31341 df-chj 31342 df-chsup 31343 df-pjh 31427 df-cv 32311 df-at 32370

This theorem is referenced by: mdsymlem3 32437

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