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Theorem nepss 31927
Description: Two classes are inequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)
Assertion
Ref Expression
nepss (𝐴𝐵 ↔ ((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) ⊊ 𝐵))

Proof of Theorem nepss
StepHypRef Expression
1 nne 2936 . . . . . 6 (¬ (𝐴𝐵) ≠ 𝐴 ↔ (𝐴𝐵) = 𝐴)
2 neeq1 2994 . . . . . . 7 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ≠ 𝐵𝐴𝐵))
32biimprcd 240 . . . . . 6 (𝐴𝐵 → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ≠ 𝐵))
41, 3syl5bi 232 . . . . 5 (𝐴𝐵 → (¬ (𝐴𝐵) ≠ 𝐴 → (𝐴𝐵) ≠ 𝐵))
54orrd 392 . . . 4 (𝐴𝐵 → ((𝐴𝐵) ≠ 𝐴 ∨ (𝐴𝐵) ≠ 𝐵))
6 inss1 3976 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
76jctl 565 . . . . 5 ((𝐴𝐵) ≠ 𝐴 → ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
8 inss2 3977 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
98jctl 565 . . . . 5 ((𝐴𝐵) ≠ 𝐵 → ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵))
107, 9orim12i 539 . . . 4 (((𝐴𝐵) ≠ 𝐴 ∨ (𝐴𝐵) ≠ 𝐵) → (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) ∨ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵)))
115, 10syl 17 . . 3 (𝐴𝐵 → (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) ∨ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵)))
12 inidm 3965 . . . . . . 7 (𝐴𝐴) = 𝐴
13 ineq2 3951 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
1412, 13syl5reqr 2809 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐵) = 𝐴)
1514necon3i 2964 . . . . 5 ((𝐴𝐵) ≠ 𝐴𝐴𝐵)
1615adantl 473 . . . 4 (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) → 𝐴𝐵)
17 ineq1 3950 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐵) = (𝐵𝐵))
18 inidm 3965 . . . . . . 7 (𝐵𝐵) = 𝐵
1917, 18syl6eq 2810 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐵) = 𝐵)
2019necon3i 2964 . . . . 5 ((𝐴𝐵) ≠ 𝐵𝐴𝐵)
2120adantl 473 . . . 4 (((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵) → 𝐴𝐵)
2216, 21jaoi 393 . . 3 ((((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) ∨ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵)) → 𝐴𝐵)
2311, 22impbii 199 . 2 (𝐴𝐵 ↔ (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) ∨ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵)))
24 df-pss 3731 . . 3 ((𝐴𝐵) ⊊ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
25 df-pss 3731 . . 3 ((𝐴𝐵) ⊊ 𝐵 ↔ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵))
2624, 25orbi12i 544 . 2 (((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) ⊊ 𝐵) ↔ (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) ∨ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵)))
2723, 26bitr4i 267 1 (𝐴𝐵 ↔ ((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) ⊊ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382  wa 383   = wceq 1632  wne 2932  cin 3714  wss 3715  wpss 3716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-in 3722  df-ss 3729  df-pss 3731
This theorem is referenced by: (None)
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