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Theorem bj-ismoored2 34416
 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 4874 . . . 4 ((𝐵𝐴𝐵 ≠ ∅) → 𝐵 𝐴)
41, 2, 3syl2anc 587 . . 3 (𝜑 𝐵 𝐴)
5 sseqin2 4167 . . 3 ( 𝐵 𝐴 ↔ ( 𝐴 𝐵) = 𝐵)
64, 5sylib 221 . 2 (𝜑 → ( 𝐴 𝐵) = 𝐵)
7 bj-ismoored.1 . . 3 (𝜑𝐴Moore)
87, 1bj-ismoored 34415 . 2 (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)
96, 8eqeltrrd 2913 1 (𝜑 𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115   ≠ wne 3007   ∩ cin 3909   ⊆ wss 3910  ∅c0 4266  ∪ cuni 4811  ∩ cint 4849  Moorecmoore 34411 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-in 3917  df-ss 3927  df-nul 4267  df-pw 4514  df-uni 4812  df-int 4850  df-bj-moore 34412 This theorem is referenced by: (None)
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