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Theorem bj-ismoored 34280
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 4877 . . . 4 (𝑥 = 𝐵 𝑥 = 𝐵)
21ineq2d 4193 . . 3 (𝑥 = 𝐵 → ( 𝐴 𝑥) = ( 𝐴 𝐵))
32eleq1d 2902 . 2 (𝑥 = 𝐵 → (( 𝐴 𝑥) ∈ 𝐴 ↔ ( 𝐴 𝐵) ∈ 𝐴))
4 bj-ismoored.1 . . 3 (𝜑𝐴Moore)
5 bj-ismoore 34278 . . 3 (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
64, 5sylib 219 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
7 bj-ismoored.2 . . 3 (𝜑𝐵𝐴)
84, 7sselpwd 5227 . 2 (𝜑𝐵 ∈ 𝒫 𝐴)
93, 6, 8rspcdva 3629 1 (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  wral 3143  cin 3939  wss 3940  𝒫 cpw 4542   cuni 4837   cint 4874  Moorecmoore 34276
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-in 3947  df-ss 3956  df-nul 4296  df-pw 4544  df-uni 4838  df-int 4875  df-bj-moore 34277
This theorem is referenced by:  bj-ismoored2  34281
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