Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ismoored2 Structured version   Visualization version   GIF version

Theorem bj-ismoored2 37091
Description: Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
Hypotheses
Ref Expression
bj-ismoored.1 (𝜑𝐴Moore)
bj-ismoored.2 (𝜑𝐵𝐴)
bj-ismoored2.3 (𝜑𝐵 ≠ ∅)
Assertion
Ref Expression
bj-ismoored2 (𝜑 𝐵𝐴)

Proof of Theorem bj-ismoored2
StepHypRef Expression
1 bj-ismoored.2 . . . 4 (𝜑𝐵𝐴)
2 bj-ismoored2.3 . . . 4 (𝜑𝐵 ≠ ∅)
3 intssuni2 4978 . . . 4 ((𝐵𝐴𝐵 ≠ ∅) → 𝐵 𝐴)
41, 2, 3syl2anc 584 . . 3 (𝜑 𝐵 𝐴)
5 sseqin2 4231 . . 3 ( 𝐵 𝐴 ↔ ( 𝐴 𝐵) = 𝐵)
64, 5sylib 218 . 2 (𝜑 → ( 𝐴 𝐵) = 𝐵)
7 bj-ismoored.1 . . 3 (𝜑𝐴Moore)
87, 1bj-ismoored 37090 . 2 (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)
96, 8eqeltrrd 2840 1 (𝜑 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wne 2938  cin 3962  wss 3963  c0 4339   cuni 4912   cint 4951  Moorecmoore 37086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-uni 4913  df-int 4952  df-bj-moore 37087
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator