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Theorem bj-ismoored 37108
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)
Assertion
Ref Expression
bj-ismoored (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)

Proof of Theorem bj-ismoored
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 inteq 4949 . . . 4 (𝑥 = 𝐵 𝑥 = 𝐵)
21ineq2d 4220 . . 3 (𝑥 = 𝐵 → ( 𝐴 𝑥) = ( 𝐴 𝐵))
32eleq1d 2826 . 2 (𝑥 = 𝐵 → (( 𝐴 𝑥) ∈ 𝐴 ↔ ( 𝐴 𝐵) ∈ 𝐴))
4 bj-ismoored.1 . . 3 (𝜑𝐴Moore)
5 bj-ismoore 37106 . . 3 (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
64, 5sylib 218 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
7 bj-ismoored.2 . . 3 (𝜑𝐵𝐴)
84, 7sselpwd 5328 . 2 (𝜑𝐵 ∈ 𝒫 𝐴)
93, 6, 8rspcdva 3623 1 (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  cin 3950  wss 3951  𝒫 cpw 4600   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-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:  bj-ismoored2  37109
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