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Theorem nepss 32941
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 (𝐴𝐵 ↔ ((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) ⊊ 𝐵))

Proof of Theorem nepss
StepHypRef Expression
1 nne 3018 . . . . . 6 (¬ (𝐴𝐵) ≠ 𝐴 ↔ (𝐴𝐵) = 𝐴)
2 neeq1 3076 . . . . . . 7 ((𝐴𝐵) = 𝐴 → ((𝐴𝐵) ≠ 𝐵𝐴𝐵))
32biimprcd 252 . . . . . 6 (𝐴𝐵 → ((𝐴𝐵) = 𝐴 → (𝐴𝐵) ≠ 𝐵))
41, 3syl5bi 244 . . . . 5 (𝐴𝐵 → (¬ (𝐴𝐵) ≠ 𝐴 → (𝐴𝐵) ≠ 𝐵))
54orrd 859 . . . 4 (𝐴𝐵 → ((𝐴𝐵) ≠ 𝐴 ∨ (𝐴𝐵) ≠ 𝐵))
6 inss1 4203 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
76jctl 526 . . . . 5 ((𝐴𝐵) ≠ 𝐴 → ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
8 inss2 4204 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
98jctl 526 . . . . 5 ((𝐴𝐵) ≠ 𝐵 → ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵))
107, 9orim12i 905 . . . 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 4193 . . . . . . 7 (𝐴𝐴) = 𝐴
13 ineq2 4181 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
1412, 13syl5reqr 2869 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐵) = 𝐴)
1514necon3i 3046 . . . . 5 ((𝐴𝐵) ≠ 𝐴𝐴𝐵)
1615adantl 484 . . . 4 (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) → 𝐴𝐵)
17 ineq1 4179 . . . . . . 7 (𝐴 = 𝐵 → (𝐴𝐵) = (𝐵𝐵))
18 inidm 4193 . . . . . . 7 (𝐵𝐵) = 𝐵
1917, 18syl6eq 2870 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐵) = 𝐵)
2019necon3i 3046 . . . . 5 ((𝐴𝐵) ≠ 𝐵𝐴𝐵)
2120adantl 484 . . . 4 (((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵) → 𝐴𝐵)
2216, 21jaoi 853 . . 3 ((((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) ∨ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵)) → 𝐴𝐵)
2311, 22impbii 211 . 2 (𝐴𝐵 ↔ (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) ∨ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵)))
24 df-pss 3952 . . 3 ((𝐴𝐵) ⊊ 𝐴 ↔ ((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴))
25 df-pss 3952 . . 3 ((𝐴𝐵) ⊊ 𝐵 ↔ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵))
2624, 25orbi12i 911 . 2 (((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) ⊊ 𝐵) ↔ (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐵) ≠ 𝐴) ∨ ((𝐴𝐵) ⊆ 𝐵 ∧ (𝐴𝐵) ≠ 𝐵)))
2723, 26bitr4i 280 1 (𝐴𝐵 ↔ ((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) ⊊ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 208  wa 398  wo 843   = wceq 1531  wne 3014  cin 3933  wss 3934  wpss 3935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-rab 3145  df-v 3495  df-in 3941  df-ss 3950  df-pss 3952
This theorem is referenced by: (None)
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