Theorem dfon2lem4 31691

Description: Lemma for dfon2 31697. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)

Hypotheses

Ref	Expression
dfon2lem4.1	⊢ 𝐴 ∈ V
dfon2lem4.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
dfon2lem4	⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem dfon2lem4

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 3833	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	sseli 3599	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐴)
3		dfon2lem4.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ V
4		dfon2lem3 31690	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ V → (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧)))
5	3, 4	ax-mp 5	. . . . . . . . . . 11 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧))
6	5	simprd 479	. . . . . . . . . 10 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧)
7		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 ∩ 𝐵) → (𝑧 ∈ 𝑧 ↔ (𝐴 ∩ 𝐵) ∈ 𝑧))
8		eleq2 2690	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 ∩ 𝐵) → ((𝐴 ∩ 𝐵) ∈ 𝑧 ↔ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵)))
9	7, 8	bitrd 268	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐴 ∩ 𝐵) → (𝑧 ∈ 𝑧 ↔ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵)))
10	9	notbid 308	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐴 ∩ 𝐵) → (¬ 𝑧 ∈ 𝑧 ↔ ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵)))
11	10	rspccv 3306	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧 → ((𝐴 ∩ 𝐵) ∈ 𝐴 → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵)))
12	6, 11	syl 17	. . . . . . . . 9 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → ((𝐴 ∩ 𝐵) ∈ 𝐴 → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵)))
13	12	adantr 481	. . . . . . . 8 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ((𝐴 ∩ 𝐵) ∈ 𝐴 → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵)))
14	2, 13	syl5 34	. . . . . . 7 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵)))
15	14	pm2.01d 181	. . . . . 6 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵))
16		elin 3796	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵))
17	15, 16	sylnib 318	. . . . 5 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ¬ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵))
18	5	simpld 475	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → Tr 𝐴)
19		dfon2lem4.2	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
20		dfon2lem3 31690	. . . . . . . . . 10 ⊢ (𝐵 ∈ V → (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → (Tr 𝐵 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝑧 ∈ 𝑧)))
21	19, 20	ax-mp 5	. . . . . . . . 9 ⊢ (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → (Tr 𝐵 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝑧 ∈ 𝑧))
22	21	simpld 475	. . . . . . . 8 ⊢ (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → Tr 𝐵)
23		trin 4763	. . . . . . . 8 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
24	18, 22, 23	syl2an 494	. . . . . . 7 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → Tr (𝐴 ∩ 𝐵))
25	3	inex1 4799	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ∈ V
26		psseq1 3694	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 ∩ 𝐵) → (𝑥 ⊊ 𝐴 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴))
27		treq 4758	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 ∩ 𝐵) → (Tr 𝑥 ↔ Tr (𝐴 ∩ 𝐵)))
28	26, 27	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ∩ 𝐵) → ((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∧ Tr (𝐴 ∩ 𝐵))))
29		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ∩ 𝐵) → (𝑥 ∈ 𝐴 ↔ (𝐴 ∩ 𝐵) ∈ 𝐴))
30	28, 29	imbi12d 334	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∩ 𝐵) → (((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ↔ (((𝐴 ∩ 𝐵) ⊊ 𝐴 ∧ Tr (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐵) ∈ 𝐴)))
31	25, 30	spcv 3299	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (((𝐴 ∩ 𝐵) ⊊ 𝐴 ∧ Tr (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐵) ∈ 𝐴))
32	31	adantr 481	. . . . . . 7 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (((𝐴 ∩ 𝐵) ⊊ 𝐴 ∧ Tr (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐵) ∈ 𝐴))
33	24, 32	mpan2d 710	. . . . . 6 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ((𝐴 ∩ 𝐵) ⊊ 𝐴 → (𝐴 ∩ 𝐵) ∈ 𝐴))
34		psseq1 3694	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 ∩ 𝐵) → (𝑦 ⊊ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐵))
35		treq 4758	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 ∩ 𝐵) → (Tr 𝑦 ↔ Tr (𝐴 ∩ 𝐵)))
36	34, 35	anbi12d 747	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ∩ 𝐵) → ((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐵 ∧ Tr (𝐴 ∩ 𝐵))))
37		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ∩ 𝐵) → (𝑦 ∈ 𝐵 ↔ (𝐴 ∩ 𝐵) ∈ 𝐵))
38	36, 37	imbi12d 334	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 ∩ 𝐵) → (((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) ↔ (((𝐴 ∩ 𝐵) ⊊ 𝐵 ∧ Tr (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐵) ∈ 𝐵)))
39	25, 38	spcv 3299	. . . . . . . 8 ⊢ (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → (((𝐴 ∩ 𝐵) ⊊ 𝐵 ∧ Tr (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐵) ∈ 𝐵))
40	39	adantl 482	. . . . . . 7 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (((𝐴 ∩ 𝐵) ⊊ 𝐵 ∧ Tr (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐵) ∈ 𝐵))
41	24, 40	mpan2d 710	. . . . . 6 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ((𝐴 ∩ 𝐵) ⊊ 𝐵 → (𝐴 ∩ 𝐵) ∈ 𝐵))
42	33, 41	anim12d 586	. . . . 5 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (((𝐴 ∩ 𝐵) ⊊ 𝐴 ∧ (𝐴 ∩ 𝐵) ⊊ 𝐵) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵)))
43	17, 42	mtod 189	. . . 4 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ¬ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∧ (𝐴 ∩ 𝐵) ⊊ 𝐵))
44		ianor 509	. . . 4 ⊢ (¬ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∧ (𝐴 ∩ 𝐵) ⊊ 𝐵) ↔ (¬ (𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ ¬ (𝐴 ∩ 𝐵) ⊊ 𝐵))
45	43, 44	sylib 208	. . 3 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (¬ (𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ ¬ (𝐴 ∩ 𝐵) ⊊ 𝐵))
46		sspss 3706	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
47	1, 46	mpbi 220	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴)
48		inss2 3834	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
49		sspss 3706	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐵 ∨ (𝐴 ∩ 𝐵) = 𝐵))
50	48, 49	mpbi 220	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐵 ∨ (𝐴 ∩ 𝐵) = 𝐵)
51		orel1 397	. . . . . 6 ⊢ (¬ (𝐴 ∩ 𝐵) ⊊ 𝐴 → (((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴) → (𝐴 ∩ 𝐵) = 𝐴))
52		orc 400	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → ((𝐴 ∩ 𝐵) = 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐵))
53	51, 52	syl6 35	. . . . 5 ⊢ (¬ (𝐴 ∩ 𝐵) ⊊ 𝐴 → (((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴) → ((𝐴 ∩ 𝐵) = 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐵)))
54		orel1 397	. . . . . 6 ⊢ (¬ (𝐴 ∩ 𝐵) ⊊ 𝐵 → (((𝐴 ∩ 𝐵) ⊊ 𝐵 ∨ (𝐴 ∩ 𝐵) = 𝐵) → (𝐴 ∩ 𝐵) = 𝐵))
55		olc 399	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = 𝐵 → ((𝐴 ∩ 𝐵) = 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐵))
56	54, 55	syl6 35	. . . . 5 ⊢ (¬ (𝐴 ∩ 𝐵) ⊊ 𝐵 → (((𝐴 ∩ 𝐵) ⊊ 𝐵 ∨ (𝐴 ∩ 𝐵) = 𝐵) → ((𝐴 ∩ 𝐵) = 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐵)))
57	53, 56	jaoa 532	. . . 4 ⊢ ((¬ (𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ ¬ (𝐴 ∩ 𝐵) ⊊ 𝐵) → ((((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊊ 𝐵 ∨ (𝐴 ∩ 𝐵) = 𝐵)) → ((𝐴 ∩ 𝐵) = 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐵)))
58	47, 50, 57	mp2ani 714	. . 3 ⊢ ((¬ (𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ ¬ (𝐴 ∩ 𝐵) ⊊ 𝐵) → ((𝐴 ∩ 𝐵) = 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐵))
59	45, 58	syl 17	. 2 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ((𝐴 ∩ 𝐵) = 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐵))
60		df-ss 3588	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
61		sseqin2 3817	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
62	60, 61	orbi12i 543	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ ((𝐴 ∩ 𝐵) = 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐵))
63	59, 62	sylibr 224	1 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574   ⊊ wpss 3575  Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-iun 4522  df-tr 4753  df-suc 5729

This theorem is referenced by:  dfon2lem5  31692