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Theorem disjpss 4172
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 3773 . . . . . . . 8 𝐵𝐵
21biantru 519 . . . . . . 7 (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐵))
3 ssin 3983 . . . . . . 7 ((𝐵𝐴𝐵𝐵) ↔ 𝐵 ⊆ (𝐴𝐵))
42, 3bitri 264 . . . . . 6 (𝐵𝐴𝐵 ⊆ (𝐴𝐵))
5 sseq2 3776 . . . . . 6 ((𝐴𝐵) = ∅ → (𝐵 ⊆ (𝐴𝐵) ↔ 𝐵 ⊆ ∅))
64, 5syl5bb 272 . . . . 5 ((𝐴𝐵) = ∅ → (𝐵𝐴𝐵 ⊆ ∅))
7 ss0 4119 . . . . 5 (𝐵 ⊆ ∅ → 𝐵 = ∅)
86, 7syl6bi 243 . . . 4 ((𝐴𝐵) = ∅ → (𝐵𝐴𝐵 = ∅))
98necon3ad 2956 . . 3 ((𝐴𝐵) = ∅ → (𝐵 ≠ ∅ → ¬ 𝐵𝐴))
109imp 393 . 2 (((𝐴𝐵) = ∅ ∧ 𝐵 ≠ ∅) → ¬ 𝐵𝐴)
11 nsspssun 4006 . . 3 𝐵𝐴𝐴 ⊊ (𝐵𝐴))
12 uncom 3908 . . . 4 (𝐵𝐴) = (𝐴𝐵)
1312psseq2i 3847 . . 3 (𝐴 ⊊ (𝐵𝐴) ↔ 𝐴 ⊊ (𝐴𝐵))
1411, 13bitri 264 . 2 𝐵𝐴𝐴 ⊊ (𝐴𝐵))
1510, 14sylib 208 1 (((𝐴𝐵) = ∅ ∧ 𝐵 ≠ ∅) → 𝐴 ⊊ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wne 2943  cun 3721  cin 3722  wss 3723  wpss 3724  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064
This theorem is referenced by:  isfin1-3  9414
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