atcvat4i - Hilbert Space Explorer

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Theorem atcvat4i 30759

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 30721	. . . . . . . 8 ⊢ (𝐴 ≠ 0_ℋ → ∃𝑥 ∈ HAtoms 𝑥 ⊆ 𝐴)
3		atelch 30706	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ HAtoms → 𝐶 ∈ C_ℋ )
4		atelch 30706	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ C_ℋ )
5		chub1 29869	. . . . . . . . . . . . . . 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 3946	. . . . . . . . . . . . . 14 ⊢ (𝐵 = 𝐶 → (𝐵 ⊆ (𝐶 ∨_ℋ 𝑥) ↔ 𝐶 ⊆ (𝐶 ∨_ℋ 𝑥)))
8	6, 7	syl5ibr 245	. . . . . . . . . . . . 13 ⊢ (𝐵 = 𝐶 → ((𝐶 ∈ HAtoms ∧ 𝑥 ∈ HAtoms) → 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))
9	8	expd 416	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐶 → (𝐶 ∈ HAtoms → (𝑥 ∈ HAtoms → 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥))))
10	9	impcom 408	. . . . . . . . . . 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 417	. . . . . . . . 9 ⊢ ((𝐶 ∈ HAtoms ∧ 𝐵 = 𝐶) → (𝑥 ∈ HAtoms → (𝑥 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
13	12	reximdvai 3200	. . . . . . . 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 413	. . . . . 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 423	. . 3 ⊢ (𝐶 ∈ HAtoms → (𝐵 = 𝐶 → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
19	18	adantl 482	. 2 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐵 = 𝐶 → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
20		atelch 30706	. . . . . . . 8 ⊢ (𝐵 ∈ HAtoms → 𝐵 ∈ C_ℋ )
21		chlejb2 29875	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐴 ∈ C_ℋ ) → (𝐶 ⊆ 𝐴 ↔ (𝐴 ∨_ℋ 𝐶) = 𝐴))
22	1, 21	mpan2 688	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ C_ℋ → (𝐶 ⊆ 𝐴 ↔ (𝐴 ∨_ℋ 𝐶) = 𝐴))
23	22	biimpa 477	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐶 ⊆ 𝐴) → (𝐴 ∨_ℋ 𝐶) = 𝐴)
24	23	sseq2d 3953	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ C_ℋ ∧ 𝐶 ⊆ 𝐴) → (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) ↔ 𝐵 ⊆ 𝐴))
25	24	biimpa 477	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ C_ℋ ∧ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ 𝐴)
26	25	expl 458	. . . . . . . . . 10 ⊢ (𝐶 ∈ C_ℋ → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ 𝐴))
27	26	adantl 482	. . . . . . . . 9 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ 𝐴))
28		chub2 29870	. . . . . . . . 9 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))
29	27, 28	jctird 527	. . . . . . . 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 483	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → 𝐵 ∈ HAtoms)
32	30, 31	jctild 526	. . . . . 6 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((𝐶 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ∈ HAtoms ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵)))))
33	32	impl 456	. . . . 5 ⊢ ((((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐵 ∈ HAtoms ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝐵))))
34		sseq1 3946	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
35		oveq2 7283	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐶 ∨_ℋ 𝑥) = (𝐶 ∨_ℋ 𝐵))
36	35	sseq2d 3953	. . . . . . 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 3561	. . . . 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 713	. . 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 420	. 2 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐶 ⊆ 𝐴 → ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
42		simpr 485	. . 3 ⊢ ((𝐴 ≠ 0_ℋ ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))
43		ioran 981	. . . 4 ⊢ (¬ (𝐵 = 𝐶 ∨ 𝐶 ⊆ 𝐴) ↔ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴))
44	1	atcvat3i 30758	. . . . . . 7 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶)) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms))
45	3	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → 𝐶 ∈ C_ℋ )
46	44	imp 407	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ∈ HAtoms)
47		simpll 764	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → 𝐵 ∈ HAtoms)
48	45, 46, 47	3jca 1127	. . . . . . . . . 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 4163	. . . . . . . . . . . . 13 ⊢ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐵 ∨_ℋ 𝐶)
50		chjcom 29868	. . . . . . . . . . . . . 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 3973	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐶 ∨_ℋ 𝐵))
53	52	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ (𝐶 ∨_ℋ 𝐵))
54		atnssm0 30738	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐶 ∈ HAtoms) → (¬ 𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 0_ℋ))
55	1, 54	mpan 687	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ HAtoms → (¬ 𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 0_ℋ))
56	55	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → (¬ 𝐶 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐶) = 0_ℋ))
57		inss1 4162	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴
58		sslin 4168	. . . . . . . . . . . . . . . . . . 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 4135	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∩ 𝐴) = (𝐴 ∩ 𝐶)
61	59, 60	sseqtri 3957	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) ⊆ (𝐴 ∩ 𝐶)
62		sseq2 3947	. . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → 𝐶 ∈ C_ℋ )
65		chjcl 29719	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ C_ℋ ∧ 𝐶 ∈ C_ℋ ) → (𝐵 ∨_ℋ 𝐶) ∈ C_ℋ )
66		chincl 29861	. . . . . . . . . . . . . . . . . . . 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 29861	. . . . . . . . . . . . . . . . . . 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 29805	. . . . . . . . . . . . . . . . 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	syl5ib 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 407	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ¬ 𝐶 ⊆ 𝐴) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ)
76	75	adantrl 713	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ (¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴)) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ)
77	76	adantrr 714	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) ∧ ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) ∧ 𝐵 ⊆ (𝐴 ∨_ℋ 𝐶))) → (𝐶 ∩ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))) = 0_ℋ)
78	53, 77	jca 512	. . . . . . . . . 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 30743	. . . . . . . . . 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 520	. . . . . . . 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 413	. . . . . . 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 513	. . . . . 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 3946	. . . . . . . 8 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝑥 ⊆ 𝐴 ↔ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) ⊆ 𝐴))
85		oveq2 7283	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∩ (𝐵 ∨_ℋ 𝐶)) → (𝐶 ∨_ℋ 𝑥) = (𝐶 ∨_ℋ (𝐴 ∩ (𝐵 ∨_ℋ 𝐶))))
86	85	sseq2d 3953	. . . . . . . 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 3561	. . . . . 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 416	. . . 4 ⊢ ((𝐵 ∈ HAtoms ∧ 𝐶 ∈ HAtoms) → ((¬ 𝐵 = 𝐶 ∧ ¬ 𝐶 ⊆ 𝐴) → (𝐵 ⊆ (𝐴 ∨_ℋ 𝐶) → ∃𝑥 ∈ HAtoms (𝑥 ⊆ 𝐴 ∧ 𝐵 ⊆ (𝐶 ∨_ℋ 𝑥)))))
91	43, 90	syl5bi 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 1031	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 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 ∩ cin 3886 ⊆ wss 3887 (class class class)co 7275 C_ℋ cch 29291 ∨_ℋ chj 29295 0_ℋc0h 29297 HAtomscat 29327

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 ax-hilex 29361 ax-hfvadd 29362 ax-hvcom 29363 ax-hvass 29364 ax-hv0cl 29365 ax-hvaddid 29366 ax-hfvmul 29367 ax-hvmulid 29368 ax-hvmulass 29369 ax-hvdistr1 29370 ax-hvdistr2 29371 ax-hvmul0 29372 ax-hfi 29441 ax-his1 29444 ax-his2 29445 ax-his3 29446 ax-his4 29447 ax-hcompl 29564

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-cn 22378 df-cnp 22379 df-lm 22380 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cfil 24419 df-cau 24420 df-cmet 24421 df-grpo 28855 df-gid 28856 df-ginv 28857 df-gdiv 28858 df-ablo 28907 df-vc 28921 df-nv 28954 df-va 28957 df-ba 28958 df-sm 28959 df-0v 28960 df-vs 28961 df-nmcv 28962 df-ims 28963 df-dip 29063 df-ssp 29084 df-ph 29175 df-cbn 29225 df-hnorm 29330 df-hba 29331 df-hvsub 29333 df-hlim 29334 df-hcau 29335 df-sh 29569 df-ch 29583 df-oc 29614 df-ch0 29615 df-shs 29670 df-span 29671 df-chj 29672 df-chsup 29673 df-pjh 29757 df-cv 30641 df-at 30700

This theorem is referenced by: mdsymlem3 30767

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