Theorem nepss 32196

Description: Two classes are unequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)

Assertion

Ref	Expression
nepss	⊢ (𝐴 ≠ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵))

Proof of Theorem nepss

Step	Hyp	Ref	Expression
1		nne 2973	. . . . . 6 ⊢ (¬ (𝐴 ∩ 𝐵) ≠ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		neeq1 3031	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → ((𝐴 ∩ 𝐵) ≠ 𝐵 ↔ 𝐴 ≠ 𝐵))
3	2	biimprcd 242	. . . . . 6 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 ∩ 𝐵) = 𝐴 → (𝐴 ∩ 𝐵) ≠ 𝐵))
4	1, 3	syl5bi 234	. . . . 5 ⊢ (𝐴 ≠ 𝐵 → (¬ (𝐴 ∩ 𝐵) ≠ 𝐴 → (𝐴 ∩ 𝐵) ≠ 𝐵))
5	4	orrd 852	. . . 4 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 ∩ 𝐵) ≠ 𝐴 ∨ (𝐴 ∩ 𝐵) ≠ 𝐵))
6		inss1 4053	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
7	6	jctl 519	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐴 → ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴))
8		inss2 4054	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
9	8	jctl 519	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐵 → ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵))
10	7, 9	orim12i 895	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ≠ 𝐴 ∨ (𝐴 ∩ 𝐵) ≠ 𝐵) → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) ∨ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵)))
11	5, 10	syl 17	. . 3 ⊢ (𝐴 ≠ 𝐵 → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) ∨ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵)))
12		inidm 4043	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
13		ineq2 4031	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐴) = (𝐴 ∩ 𝐵))
14	12, 13	syl5reqr 2829	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
15	14	necon3i 3001	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐴 → 𝐴 ≠ 𝐵)
16	15	adantl 475	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) → 𝐴 ≠ 𝐵)
17		ineq1 4030	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐵))
18		inidm 4043	. . . . . . 7 ⊢ (𝐵 ∩ 𝐵) = 𝐵
19	17, 18	syl6eq 2830	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵)
20	19	necon3i 3001	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐵 → 𝐴 ≠ 𝐵)
21	20	adantl 475	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵) → 𝐴 ≠ 𝐵)
22	16, 21	jaoi 846	. . 3 ⊢ ((((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) ∨ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵)) → 𝐴 ≠ 𝐵)
23	11, 22	impbii 201	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) ∨ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵)))
24		df-pss 3808	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴))
25		df-pss 3808	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵))
26	24, 25	orbi12i 901	. 2 ⊢ (((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) ∨ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵)))
27	23, 26	bitr4i 270	1 ⊢ (𝐴 ≠ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 198   ∧ wa 386   ∨ wo 836   = wceq 1601   ≠ wne 2969   ∩ cin 3791   ⊆ wss 3792   ⊊ wpss 3793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-v 3400  df-in 3799  df-ss 3806  df-pss 3808

This theorem is referenced by: (None)