Theorem nepss 34675

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 2944	. . . . . 6 ⊢ (¬ (𝐴 ∩ 𝐵) ≠ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		neeq1 3003	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → ((𝐴 ∩ 𝐵) ≠ 𝐵 ↔ 𝐴 ≠ 𝐵))
3	2	biimprcd 249	. . . . . 6 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 ∩ 𝐵) = 𝐴 → (𝐴 ∩ 𝐵) ≠ 𝐵))
4	1, 3	biimtrid 241	. . . . 5 ⊢ (𝐴 ≠ 𝐵 → (¬ (𝐴 ∩ 𝐵) ≠ 𝐴 → (𝐴 ∩ 𝐵) ≠ 𝐵))
5	4	orrd 861	. . . 4 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 ∩ 𝐵) ≠ 𝐴 ∨ (𝐴 ∩ 𝐵) ≠ 𝐵))
6		inss1 4227	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
7	6	jctl 524	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐴 → ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴))
8		inss2 4228	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
9	8	jctl 524	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐵 → ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵))
10	7, 9	orim12i 907	. . . 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		ineq2 4205	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐴) = (𝐴 ∩ 𝐵))
13		inidm 4217	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
14	12, 13	eqtr3di 2787	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
15	14	necon3i 2973	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐴 → 𝐴 ≠ 𝐵)
16	15	adantl 482	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) → 𝐴 ≠ 𝐵)
17		ineq1 4204	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐵))
18		inidm 4217	. . . . . . 7 ⊢ (𝐵 ∩ 𝐵) = 𝐵
19	17, 18	eqtrdi 2788	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵)
20	19	necon3i 2973	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐵 → 𝐴 ≠ 𝐵)
21	20	adantl 482	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵) → 𝐴 ≠ 𝐵)
22	16, 21	jaoi 855	. . 3 ⊢ ((((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) ∨ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵)) → 𝐴 ≠ 𝐵)
23	11, 22	impbii 208	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) ∨ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵)))
24		df-pss 3966	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴))
25		df-pss 3966	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵))
26	24, 25	orbi12i 913	. 2 ⊢ (((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) ∨ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵)))
27	23, 26	bitr4i 277	1 ⊢ (𝐴 ≠ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ≠ wne 2940   ∩ cin 3946   ⊆ wss 3947   ⊊ wpss 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-in 3954  df-ss 3964  df-pss 3966

This theorem is referenced by: (None)