Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjpss Structured version   Visualization version   GIF version

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 3765 . . . . . . . 8 𝐵𝐵
21biantru 527 . . . . . . 7 (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐵))
3 ssin 3978 . . . . . . 7 ((𝐵𝐴𝐵𝐵) ↔ 𝐵 ⊆ (𝐴𝐵))
42, 3bitri 264 . . . . . 6 (𝐵𝐴𝐵 ⊆ (𝐴𝐵))
5 sseq2 3768 . . . . . 6 ((𝐴𝐵) = ∅ → (𝐵 ⊆ (𝐴𝐵) ↔ 𝐵 ⊆ ∅))
64, 5syl5bb 272 . . . . 5 ((𝐴𝐵) = ∅ → (𝐵𝐴𝐵 ⊆ ∅))
7 ss0 4117 . . . . 5 (𝐵 ⊆ ∅ → 𝐵 = ∅)
86, 7syl6bi 243 . . . 4 ((𝐴𝐵) = ∅ → (𝐵𝐴𝐵 = ∅))
98necon3ad 2945 . . 3 ((𝐴𝐵) = ∅ → (𝐵 ≠ ∅ → ¬ 𝐵𝐴))
109imp 444 . 2 (((𝐴𝐵) = ∅ ∧ 𝐵 ≠ ∅) → ¬ 𝐵𝐴)
11 nsspssun 4000 . . 3 𝐵𝐴𝐴 ⊊ (𝐵𝐴))
12 uncom 3900 . . . 4 (𝐵𝐴) = (𝐴𝐵)
1312psseq2i 3839 . . 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 383   = wceq 1632   ≠ wne 2932   ∪ cun 3713   ∩ cin 3714   ⊆ wss 3715   ⊊ wpss 3716  ∅c0 4058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059 This theorem is referenced by:  isfin1-3  9400
 Copyright terms: Public domain W3C validator