Theorem nepss 35953

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 2939	. . . . . 6 ⊢ (¬ (𝐴 ∩ 𝐵) ≠ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		neeq1 2997	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → ((𝐴 ∩ 𝐵) ≠ 𝐵 ↔ 𝐴 ≠ 𝐵))
3	2	biimprcd 251	. . . . . 6 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 ∩ 𝐵) = 𝐴 → (𝐴 ∩ 𝐵) ≠ 𝐵))
4	1, 3	biimtrid 243	. . . . 5 ⊢ (𝐴 ≠ 𝐵 → (¬ (𝐴 ∩ 𝐵) ≠ 𝐴 → (𝐴 ∩ 𝐵) ≠ 𝐵))
5	4	orrd 869	. . . 4 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 ∩ 𝐵) ≠ 𝐴 ∨ (𝐴 ∩ 𝐵) ≠ 𝐵))
6		inss1 4172	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
7	6	jctl 528	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐴 → ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴))
8		inss2 4173	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
9	8	jctl 528	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐵 → ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵))
10	7, 9	orim12i 914	. . . 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 4150	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐴) = (𝐴 ∩ 𝐵))
13		inidm 4162	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
14	12, 13	eqtr3di 2790	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐴)
15	14	necon3i 2967	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐴 → 𝐴 ≠ 𝐵)
16	15	adantl 482	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) → 𝐴 ≠ 𝐵)
17		ineq1 4149	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐵))
18		inidm 4162	. . . . . . 7 ⊢ (𝐵 ∩ 𝐵) = 𝐵
19	17, 18	eqtrdi 2791	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = 𝐵)
20	19	necon3i 2967	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ≠ 𝐵 → 𝐴 ≠ 𝐵)
21	20	adantl 482	. . . 4 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵) → 𝐴 ≠ 𝐵)
22	16, 21	jaoi 863	. . 3 ⊢ ((((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) ∨ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵)) → 𝐴 ≠ 𝐵)
23	11, 22	impbii 210	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) ∨ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵)))
24		df-pss 3910	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴))
25		df-pss 3910	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊊ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵))
26	24, 25	orbi12i 920	. 2 ⊢ (((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐵) ≠ 𝐴) ∨ ((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ (𝐴 ∩ 𝐵) ≠ 𝐵)))
27	23, 26	bitr4i 279	1 ⊢ (𝐴 ≠ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ≠ wne 2935   ∩ cin 3889   ⊆ wss 3890   ⊊ wpss 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-rab 3393  df-v 3434  df-in 3897  df-ss 3907  df-pss 3910

This theorem is referenced by: (None)