Theorem dfon2lem4 35809

Description: Lemma for dfon2 35815. 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 4217	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	sseli 3959	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐴)
3		dfon2lem4.1	. . . . . . . . . . . 12 ⊢ 𝐴 ∈ V
4		dfon2lem3 35808	. . . . . . . . . . . 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 495	. . . . . . . . . 10 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧)
7		eleq1 2823	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 ∩ 𝐵) → (𝑧 ∈ 𝑧 ↔ (𝐴 ∩ 𝐵) ∈ 𝑧))
8		eleq2 2824	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐴 ∩ 𝐵) → ((𝐴 ∩ 𝐵) ∈ 𝑧 ↔ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵)))
9	7, 8	bitrd 279	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐴 ∩ 𝐵) → (𝑧 ∈ 𝑧 ↔ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵)))
10	9	notbid 318	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐴 ∩ 𝐵) → (¬ 𝑧 ∈ 𝑧 ↔ ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵)))
11	10	rspccv 3603	. . . . . . . . . 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 480	. . . . . . . 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 190	. . . . . 6 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵))
16		elin 3947	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵))
17	15, 16	sylnib 328	. . . . 5 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ¬ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵))
18	5	simpld 494	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → Tr 𝐴)
19		dfon2lem4.2	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
20		dfon2lem3 35808	. . . . . . . . . 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 494	. . . . . . . 8 ⊢ (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → Tr 𝐵)
23		trin 5246	. . . . . . . 8 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))
24	18, 22, 23	syl2an 596	. . . . . . 7 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → Tr (𝐴 ∩ 𝐵))
25	3	inex1 5292	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ∈ V
26		psseq1 4070	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 ∩ 𝐵) → (𝑥 ⊊ 𝐴 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐴))
27		treq 5242	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 ∩ 𝐵) → (Tr 𝑥 ↔ Tr (𝐴 ∩ 𝐵)))
28	26, 27	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ∩ 𝐵) → ((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∧ Tr (𝐴 ∩ 𝐵))))
29		eleq1 2823	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ∩ 𝐵) → (𝑥 ∈ 𝐴 ↔ (𝐴 ∩ 𝐵) ∈ 𝐴))
30	28, 29	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 ∩ 𝐵) → (((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ↔ (((𝐴 ∩ 𝐵) ⊊ 𝐴 ∧ Tr (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐵) ∈ 𝐴)))
31	25, 30	spcv 3589	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (((𝐴 ∩ 𝐵) ⊊ 𝐴 ∧ Tr (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐵) ∈ 𝐴))
32	31	adantr 480	. . . . . . 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 694	. . . . . 6 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ((𝐴 ∩ 𝐵) ⊊ 𝐴 → (𝐴 ∩ 𝐵) ∈ 𝐴))
34		psseq1 4070	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 ∩ 𝐵) → (𝑦 ⊊ 𝐵 ↔ (𝐴 ∩ 𝐵) ⊊ 𝐵))
35		treq 5242	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐴 ∩ 𝐵) → (Tr 𝑦 ↔ Tr (𝐴 ∩ 𝐵)))
36	34, 35	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ∩ 𝐵) → ((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐵 ∧ Tr (𝐴 ∩ 𝐵))))
37		eleq1 2823	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 ∩ 𝐵) → (𝑦 ∈ 𝐵 ↔ (𝐴 ∩ 𝐵) ∈ 𝐵))
38	36, 37	imbi12d 344	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 ∩ 𝐵) → (((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) ↔ (((𝐴 ∩ 𝐵) ⊊ 𝐵 ∧ Tr (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐵) ∈ 𝐵)))
39	25, 38	spcv 3589	. . . . . . . 8 ⊢ (∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵) → (((𝐴 ∩ 𝐵) ⊊ 𝐵 ∧ Tr (𝐴 ∩ 𝐵)) → (𝐴 ∩ 𝐵) ∈ 𝐵))
40	39	adantl 481	. . . . . . 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 694	. . . . . 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 609	. . . . 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 198	. . . 4 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → ¬ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∧ (𝐴 ∩ 𝐵) ⊊ 𝐵))
44		ianor 983	. . . 4 ⊢ (¬ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∧ (𝐴 ∩ 𝐵) ⊊ 𝐵) ↔ (¬ (𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ ¬ (𝐴 ∩ 𝐵) ⊊ 𝐵))
45	43, 44	sylib 218	. . 3 ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (¬ (𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ ¬ (𝐴 ∩ 𝐵) ⊊ 𝐵))
46		sspss 4082	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
47	1, 46	mpbi 230	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴)
48		inss2 4218	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
49		sspss 4082	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐵 ∨ (𝐴 ∩ 𝐵) = 𝐵))
50	48, 49	mpbi 230	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐵 ∨ (𝐴 ∩ 𝐵) = 𝐵)
51		orel1 888	. . . . . 6 ⊢ (¬ (𝐴 ∩ 𝐵) ⊊ 𝐴 → (((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴) → (𝐴 ∩ 𝐵) = 𝐴))
52		orc 867	. . . . . 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 888	. . . . . 6 ⊢ (¬ (𝐴 ∩ 𝐵) ⊊ 𝐵 → (((𝐴 ∩ 𝐵) ⊊ 𝐵 ∨ (𝐴 ∩ 𝐵) = 𝐵) → (𝐴 ∩ 𝐵) = 𝐵))
55		olc 868	. . . . . 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 957	. . . 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 698	. . 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		dfss2 3949	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
61		sseqin2 4203	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵)
62	60, 61	orbi12i 914	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ ((𝐴 ∩ 𝐵) = 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐵))
63	59, 62	sylibr 234	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   ∧ wa 395   ∨ wo 847  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931   ⊊ wpss 3932  Tr wtr 5234

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-pw 4582  df-sn 4607  df-pr 4609  df-uni 4889  df-iun 4974  df-tr 5235  df-suc 6363

This theorem is referenced by:  dfon2lem5  35810