atcvat4i - Hilbert Space Explorer

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Theorem atcvat4i 32416

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 32378	. . . . . . . 8 ⊢ (𝐴 ≠ 0_ℋ → ∃𝑥 ∈ HAtoms 𝑥 ⊆ 𝐴)
3		atelch 32363	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ HAtoms → 𝐶 ∈ C_ℋ )
4		atelch 32363	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ C_ℋ )
5		chub1 31526	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → 𝐶 ⊆ (𝐶 ∨_ℋ 𝑥))
6	3, 4, 5	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ HAtoms ∧ 𝑥 ∈ HAtoms) → 𝐶 ⊆ (𝐶 ∨_ℋ 𝑥))
7		sseq1 4009	. . . . . . . . . . . . . 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 612	. . . . . . . . . 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 3165	. . . . . . . 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 32363	. . . . . . . 8 ⊢ (𝐵 ∈ HAtoms → 𝐵 ∈ C_ℋ )
21		chlejb2 31532	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → (𝐶 ⊆ 𝐴 ↔ (𝐴 ∨_ℋ 𝐶) = 𝐴))
22	1, 21	mpan2 691	. . . . . . . . . . . . . 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 4016	. . . . . . . . . . . 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 31527	. . . . . . . . 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 596	. . . . . . 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 4009	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
35		oveq2 7439	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐶 ∨_ℋ 𝑥) = (𝐶 ∨_ℋ 𝐵))
36	35	sseq2d 4016	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐵 ⊆ (𝐶 ∨_ℋ 𝑥) ↔ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵)))
37	34, 36	anbi12d 632	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
38	37	rspcev 3622	. . . . 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 716	. . 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 986	. . . 4 ⊢ (¬ (𝐵 = 𝐶 ∨ 𝐶 ⊆ 𝐴) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴))
44	1	atcvat3i 32415	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms))
45	3	ad2antlr 727	. . . . . . . . . . 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 767	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → 𝐵 ∈ HAtoms)
48	45, 46, 47	3jca 1129	. . . . . . . . . 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 4238	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐵 ∨_ℋ 𝐶)
50		chjcom 31525	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝐶) = (𝐶 ∨_ℋ 𝐵))
51	20, 3, 50	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐵 ∨_ℋ 𝐶) = (𝐶 ∨_ℋ 𝐵))
52	49, 51	sseqtrid 4026	. . . . . . . . . . . 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 32395	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐶 ∈ HAtoms) → (¬ 𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 0_ℋ))
55	1, 54	mpan 690	. . . . . . . . . . . . . . . 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 4237	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴
58		sslin 4243	. . . . . . . . . . . . . . . . . . 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 4209	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∩ 𝐴) = (𝐴 ∩ 𝐶)
61	59, 60	sseqtri 4032	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ (𝐴 ∩ 𝐶)
62		sseq2 4010	. . . . . . . . . . . . . . . . 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 31376	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝐶) ∈ C_ℋ )
66		chincl 31518	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ C_ℋ ∧ (𝐵 ∨_ℋ 𝐶) ∈ C_ℋ ) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ C_ℋ )
67	1, 65, 66	sylancr 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ C_ℋ )
68		chincl 31518	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ C_ℋ ∧ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ C_ℋ ) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ )
69	64, 67, 68	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ )
70	20, 3, 69	syl2an 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ )
71		chle0 31462	. . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴)) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ)
77	76	adantrr 717	. . . . . . . . . . 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 32400	. . . . . . . . . 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 4009	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝑥 ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴))
85		oveq2 7439	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝐶 ∨_ℋ 𝑥) = (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))
86	85	sseq2d 4016	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝐵 ⊆ (𝐶 ∨_ℋ 𝑥) ↔ 𝐵 ⊆ (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)))))
87	84, 86	anbi12d 632	. . . . . . 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 3622	. . . . . 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 1035	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 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ∩ cin 3950 ⊆ wss 3951 (class class class)co 7431 C_ℋ cch 30948 ∨_ℋ chj 30952 0_ℋc0h 30954 HAtomscat 30984

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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 ax-hilex 31018 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvmulass 31026 ax-hvdistr1 31027 ax-hvdistr2 31028 ax-hvmul0 31029 ax-hfi 31098 ax-his1 31101 ax-his2 31102 ax-his3 31103 ax-his4 31104 ax-hcompl 31221

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-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 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-pred 6321 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-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-cn 23235 df-cnp 23236 df-lm 23237 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cfil 25289 df-cau 25290 df-cmet 25291 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-vs 30618 df-nmcv 30619 df-ims 30620 df-dip 30720 df-ssp 30741 df-ph 30832 df-cbn 30882 df-hnorm 30987 df-hba 30988 df-hvsub 30990 df-hlim 30991 df-hcau 30992 df-sh 31226 df-ch 31240 df-oc 31271 df-ch0 31272 df-shs 31327 df-span 31328 df-chj 31329 df-chsup 31330 df-pjh 31414 df-cv 32298 df-at 32357

This theorem is referenced by: mdsymlem3 32424

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