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Theorem disjpss 4408
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 3987 . . . . . . . 8 𝐵𝐵
21biantru 532 . . . . . . 7 (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐵))
3 ssin 4205 . . . . . . 7 ((𝐵𝐴𝐵𝐵) ↔ 𝐵 ⊆ (𝐴𝐵))
42, 3bitri 277 . . . . . 6 (𝐵𝐴𝐵 ⊆ (𝐴𝐵))
5 sseq2 3991 . . . . . 6 ((𝐴𝐵) = ∅ → (𝐵 ⊆ (𝐴𝐵) ↔ 𝐵 ⊆ ∅))
64, 5syl5bb 285 . . . . 5 ((𝐴𝐵) = ∅ → (𝐵𝐴𝐵 ⊆ ∅))
7 ss0 4350 . . . . 5 (𝐵 ⊆ ∅ → 𝐵 = ∅)
86, 7syl6bi 255 . . . 4 ((𝐴𝐵) = ∅ → (𝐵𝐴𝐵 = ∅))
98necon3ad 3027 . . 3 ((𝐴𝐵) = ∅ → (𝐵 ≠ ∅ → ¬ 𝐵𝐴))
109imp 409 . 2 (((𝐴𝐵) = ∅ ∧ 𝐵 ≠ ∅) → ¬ 𝐵𝐴)
11 nsspssun 4232 . . 3 𝐵𝐴𝐴 ⊊ (𝐵𝐴))
12 uncom 4127 . . . 4 (𝐵𝐴) = (𝐴𝐵)
1312psseq2i 4065 . . 3 (𝐴 ⊊ (𝐵𝐴) ↔ 𝐴 ⊊ (𝐴𝐵))
1411, 13bitri 277 . 2 𝐵𝐴𝐴 ⊊ (𝐴𝐵))
1510, 14sylib 220 1 (((𝐴𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1530  wne 3014  cun 3932  cin 3933  wss 3934  wpss 3935  c0 4289
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290
This theorem is referenced by:  omsucne  7590  isfin1-3  9800
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