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Theorem bj-ismoored2 33644
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 4737 . . . 4 ((𝐵𝐴𝐵 ≠ ∅) → 𝐵 𝐴)
41, 2, 3syl2anc 579 . . 3 (𝜑 𝐵 𝐴)
5 sseqin2 4040 . . 3 ( 𝐵 𝐴 ↔ ( 𝐴 𝐵) = 𝐵)
64, 5sylib 210 . 2 (𝜑 → ( 𝐴 𝐵) = 𝐵)
7 bj-ismoored.1 . . 3 (𝜑𝐴Moore)
87, 1bj-ismoored 33643 . 2 (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)
96, 8eqeltrrd 2860 1 (𝜑 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  wne 2969  cin 3791  wss 3792  c0 4141   cuni 4673   cint 4712  Moorecmoore 33638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754  ax-sep 5019
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-v 3400  df-dif 3795  df-in 3799  df-ss 3806  df-nul 4142  df-pw 4381  df-uni 4674  df-int 4713  df-bj-moore 33639
This theorem is referenced by: (None)
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