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 37109
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 4973 . . . 4 ((𝐵𝐴𝐵 ≠ ∅) → 𝐵 𝐴)
41, 2, 3syl2anc 584 . . 3 (𝜑 𝐵 𝐴)
5 sseqin2 4223 . . 3 ( 𝐵 𝐴 ↔ ( 𝐴 𝐵) = 𝐵)
64, 5sylib 218 . 2 (𝜑 → ( 𝐴 𝐵) = 𝐵)
7 bj-ismoored.1 . . 3 (𝜑𝐴Moore)
87, 1bj-ismoored 37108 . 2 (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)
96, 8eqeltrrd 2842 1 (𝜑 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wne 2940  cin 3950  wss 3951  c0 4333   cuni 4907   cint 4946  Moorecmoore 37104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602  df-uni 4908  df-int 4947  df-bj-moore 37105
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator