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Theorem bj-ismoored2 36640
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 4972 . . . 4 ((𝐵𝐴𝐵 ≠ ∅) → 𝐵 𝐴)
41, 2, 3syl2anc 582 . . 3 (𝜑 𝐵 𝐴)
5 sseqin2 4210 . . 3 ( 𝐵 𝐴 ↔ ( 𝐴 𝐵) = 𝐵)
64, 5sylib 217 . 2 (𝜑 → ( 𝐴 𝐵) = 𝐵)
7 bj-ismoored.1 . . 3 (𝜑𝐴Moore)
87, 1bj-ismoored 36639 . 2 (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)
96, 8eqeltrrd 2826 1 (𝜑 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wne 2930  cin 3940  wss 3941  c0 4319   cuni 4904   cint 4945  Moorecmoore 36635
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-in 3948  df-ss 3958  df-nul 4320  df-pw 4601  df-uni 4905  df-int 4946  df-bj-moore 36636
This theorem is referenced by: (None)
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