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Theorem disjpss 4453
Description: A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
Assertion
Ref Expression
disjpss (((𝐴𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴𝐵))

Proof of Theorem disjpss
StepHypRef Expression
1 ssid 3997 . . . . . . . 8 𝐵𝐵
21biantru 529 . . . . . . 7 (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐵))
3 ssin 4223 . . . . . . 7 ((𝐵𝐴𝐵𝐵) ↔ 𝐵 ⊆ (𝐴𝐵))
42, 3bitri 275 . . . . . 6 (𝐵𝐴𝐵 ⊆ (𝐴𝐵))
5 sseq2 4001 . . . . . 6 ((𝐴𝐵) = ∅ → (𝐵 ⊆ (𝐴𝐵) ↔ 𝐵 ⊆ ∅))
64, 5bitrid 283 . . . . 5 ((𝐴𝐵) = ∅ → (𝐵𝐴𝐵 ⊆ ∅))
7 ss0 4391 . . . . 5 (𝐵 ⊆ ∅ → 𝐵 = ∅)
86, 7biimtrdi 252 . . . 4 ((𝐴𝐵) = ∅ → (𝐵𝐴𝐵 = ∅))
98necon3ad 2945 . . 3 ((𝐴𝐵) = ∅ → (𝐵 ≠ ∅ → ¬ 𝐵𝐴))
109imp 406 . 2 (((𝐴𝐵) = ∅ ∧ 𝐵 ≠ ∅) → ¬ 𝐵𝐴)
11 nsspssun 4250 . . 3 𝐵𝐴𝐴 ⊊ (𝐵𝐴))
12 uncom 4146 . . . 4 (𝐵𝐴) = (𝐴𝐵)
1312psseq2i 4083 . . 3 (𝐴 ⊊ (𝐵𝐴) ↔ 𝐴 ⊊ (𝐴𝐵))
1411, 13bitri 275 . 2 𝐵𝐴𝐴 ⊊ (𝐴𝐵))
1510, 14sylib 217 1 (((𝐴𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wne 2932  cun 3939  cin 3940  wss 3941  wpss 3942  c0 4315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316
This theorem is referenced by:  omsucne  7868  isfin1-3  10378
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