Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nepss Structured version   Visualization version   GIF version

Theorem nepss 30854
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 2786 . . . . . 6 (¬ (𝐴𝐵) ≠ 𝐴 ↔ (𝐴𝐵) = 𝐴)
2 neeq1 2844 . . . . . . 7 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ≠ 𝐵𝐴𝐵))
32biimprcd 239 . . . . . 6 (𝐴𝐵 → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ≠ 𝐵))
41, 3syl5bi 231 . . . . 5 (𝐴𝐵 → (¬ (𝐴𝐵) ≠ 𝐴 → (𝐴𝐵) ≠ 𝐵))
54orrd 392 . . . 4 (𝐴𝐵 → ((𝐴𝐵) ≠ 𝐴 ∨ (𝐴𝐵) ≠ 𝐵))
6 inss1 3795 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
76jctl 562 . . . . 5 ((𝐴𝐵) ≠ 𝐴 → ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
8 inss2 3796 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
98jctl 562 . . . . 5 ((𝐴𝐵) ≠ 𝐵 → ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵))
107, 9orim12i 537 . . . 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 3784 . . . . . . 7 (𝐴𝐴) = 𝐴
13 ineq2 3770 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
1412, 13syl5reqr 2659 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐵) = 𝐴)
1514necon3i 2814 . . . . 5 ((𝐴𝐵) ≠ 𝐴𝐴𝐵)
1615adantl 481 . . . 4 (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) → 𝐴𝐵)
17 ineq1 3769 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐵) = (𝐵𝐵))
18 inidm 3784 . . . . . . 7 (𝐵𝐵) = 𝐵
1917, 18syl6eq 2660 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐵) = 𝐵)
2019necon3i 2814 . . . . 5 ((𝐴𝐵) ≠ 𝐵𝐴𝐵)
2120adantl 481 . . . 4 (((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵) → 𝐴𝐵)
2216, 21jaoi 393 . . 3 ((((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) ∨ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵)) → 𝐴𝐵)
2311, 22impbii 198 . 2 (𝐴𝐵 ↔ (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) ∨ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵)))
24 df-pss 3556 . . 3 ((𝐴𝐵) ⊊ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
25 df-pss 3556 . . 3 ((𝐴𝐵) ⊊ 𝐵 ↔ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵))
2624, 25orbi12i 542 . 2 (((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) ⊊ 𝐵) ↔ (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) ∨ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵)))
2723, 26bitr4i 266 1 (𝐴𝐵 ↔ ((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) ⊊ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wo 382  wa 383   = wceq 1475  wne 2780  cin 3539  wss 3540  wpss 3541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-in 3547  df-ss 3554  df-pss 3556
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator