atcvat4i - Hilbert Space Explorer

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Theorem atcvat4i 31650

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 31612	. . . . . . . 8 ⊢ (𝐴 ≠ 0_ℋ → ∃𝑥 ∈ HAtoms 𝑥 ⊆ 𝐴)
3		atelch 31597	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ HAtoms → 𝐶 ∈ C_ℋ )
4		atelch 31597	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ C_ℋ )
5		chub1 30760	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝑥 ∈ C_ℋ ) → 𝐶 ⊆ (𝐶 ∨_ℋ 𝑥))
6	3, 4, 5	syl2an 597	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ HAtoms ∧ 𝑥 ∈ HAtoms) → 𝐶 ⊆ (𝐶 ∨_ℋ 𝑥))
7		sseq1 4008	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝐶 → (𝐵 ⊆ (𝐶 ∨_ℋ 𝑥) ↔ 𝐶 ⊆ (𝐶 ∨_ℋ 𝑥)))
8	6, 7	imbitrrid 245	. . . . . . . . . . . . 13 ⊢ (𝐵 = 𝐶 → ((𝐶 ∈ HAtoms ∧ 𝑥 ∈ HAtoms) → 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
9	8	expd 417	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐶 → (𝐶 ∈ HAtoms → (𝑥 ∈ HAtoms → 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
10	9	impcom 409	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → (𝑥 ∈ HAtoms → 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
11	10	anim2d 613	. . . . . . . . . 10 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ∈ HAtoms) → (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
12	11	expcomd 418	. . . . . . . . 9 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → (𝑥 ∈ HAtoms → (𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
13	12	reximdvai 3166	. . . . . . . 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 414	. . . . . 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 424	. . 3 ⊢ (𝐶 ∈ HAtoms → (𝐵 = 𝐶 → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
19	18	adantl 483	. 2 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐵 = 𝐶 → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
20		atelch 31597	. . . . . . . 8 ⊢ (𝐵 ∈ HAtoms → 𝐵 ∈ C_ℋ )
21		chlejb2 30766	. . . . . . . . . . . . . . 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 478	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐶 ⊆ 𝐴) → (𝐴 ∨_ℋ 𝐶) = 𝐴)
24	23	sseq2d 4015	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐶 ⊆ 𝐴) → (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) ↔ 𝐵 ⊆ 𝐴))
25	24	biimpa 478	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ C_ℋ ∧ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ 𝐴)
26	25	expl 459	. . . . . . . . . 10 ⊢ (𝐶 ∈ C_ℋ → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ 𝐴))
27	26	adantl 483	. . . . . . . . 9 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ 𝐴))
28		chub2 30761	. . . . . . . . 9 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))
29	27, 28	jctird 528	. . . . . . . 8 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
30	20, 3, 29	syl2an 597	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
31		simpl 484	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → 𝐵 ∈ HAtoms)
32	30, 31	jctild 527	. . . . . 6 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ∈ HAtoms ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵)))))
33	32	impl 457	. . . . 5 ⊢ ((((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ∈ HAtoms ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
34		sseq1 4008	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
35		oveq2 7417	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐶 ∨_ℋ 𝑥) = (𝐶 ∨_ℋ 𝐵))
36	35	sseq2d 4015	. . . . . . 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 3613	. . . . 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 421	. 2 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐶 ⊆ 𝐴 → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
42		simpr 486	. . 3 ⊢ ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))
43		ioran 983	. . . 4 ⊢ (¬ (𝐵 = 𝐶 ∨ 𝐶 ⊆ 𝐴) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴))
44	1	atcvat3i 31649	. . . . . . 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 408	. . . . . . . . . . 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 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 4230	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐵 ∨_ℋ 𝐶)
50		chjcom 30759	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝐶) = (𝐶 ∨_ℋ 𝐵))
51	20, 3, 50	syl2an 597	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐵 ∨_ℋ 𝐶) = (𝐶 ∨_ℋ 𝐵))
52	49, 51	sseqtrid 4035	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐶 ∨_ℋ 𝐵))
53	52	adantr 482	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐶 ∨_ℋ 𝐵))
54		atnssm0 31629	. . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (¬ 𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 0_ℋ))
57		inss1 4229	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴
58		sslin 4235	. . . . . . . . . . . . . . . . . . 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 4202	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∩ 𝐴) = (𝐴 ∩ 𝐶)
61	59, 60	sseqtri 4019	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ (𝐴 ∩ 𝐶)
62		sseq2 4009	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∩ 𝐶) = 0_ℋ → ((𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ (𝐴 ∩ 𝐶) ↔ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ 0_ℋ))
63	61, 62	mpbii 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝐶) = 0_ℋ → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ 0_ℋ)
64		simpr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → 𝐶 ∈ C_ℋ )
65		chjcl 30610	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝐶) ∈ C_ℋ )
66		chincl 30752	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ C_ℋ ∧ (𝐵 ∨_ℋ 𝐶) ∈ C_ℋ ) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ C_ℋ )
67	1, 65, 66	sylancr 588	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ C_ℋ )
68		chincl 30752	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ C_ℋ ∧ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ C_ℋ ) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ )
69	64, 67, 68	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ )
70	20, 3, 69	syl2an 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ∈ C_ℋ )
71		chle0 30696	. . . . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐴 ∩ 𝐶) = 0_ℋ → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ))
74	56, 73	sylbid 239	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (¬ 𝐶 ⊆ 𝐴 → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ))
75	74	imp 408	. . . . . . . . . . . . 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 513	. . . . . . . . . 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 31634	. . . . . . . . . 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 521	. . . . . . . 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 414	. . . . . . 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 514	. . . . . 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 4008	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝑥 ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴))
85		oveq2 7417	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝐶 ∨_ℋ 𝑥) = (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))
86	85	sseq2d 4015	. . . . . . . 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 3613	. . . . . 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 417	. . . 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 241	. . 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 1033	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 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 ∩ cin 3948 ⊆ wss 3949 (class class class)co 7409 C_ℋ cch 30182 ∨_ℋ chj 30186 0_ℋc0h 30188 HAtomscat 30218

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 ax-hilex 30252 ax-hfvadd 30253 ax-hvcom 30254 ax-hvass 30255 ax-hv0cl 30256 ax-hvaddid 30257 ax-hfvmul 30258 ax-hvmulid 30259 ax-hvmulass 30260 ax-hvdistr1 30261 ax-hvdistr2 30262 ax-hvmul0 30263 ax-hfi 30332 ax-his1 30335 ax-his2 30336 ax-his3 30337 ax-his4 30338 ax-hcompl 30455

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-cn 22731 df-cnp 22732 df-lm 22733 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cfil 24772 df-cau 24773 df-cmet 24774 df-grpo 29746 df-gid 29747 df-ginv 29748 df-gdiv 29749 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-vs 29852 df-nmcv 29853 df-ims 29854 df-dip 29954 df-ssp 29975 df-ph 30066 df-cbn 30116 df-hnorm 30221 df-hba 30222 df-hvsub 30224 df-hlim 30225 df-hcau 30226 df-sh 30460 df-ch 30474 df-oc 30505 df-ch0 30506 df-shs 30561 df-span 30562 df-chj 30563 df-chsup 30564 df-pjh 30648 df-cv 31532 df-at 31591

This theorem is referenced by: mdsymlem3 31658

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