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Theorem bj-ismoored 34795
 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 4842 . . . 4 (𝑥 = 𝐵 𝑥 = 𝐵)
21ineq2d 4118 . . 3 (𝑥 = 𝐵 → ( 𝐴 𝑥) = ( 𝐴 𝐵))
32eleq1d 2837 . 2 (𝑥 = 𝐵 → (( 𝐴 𝑥) ∈ 𝐴 ↔ ( 𝐴 𝐵) ∈ 𝐴))
4 bj-ismoored.1 . . 3 (𝜑𝐴Moore)
5 bj-ismoore 34793 . . 3 (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
64, 5sylib 221 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
7 bj-ismoored.2 . . 3 (𝜑𝐵𝐴)
84, 7sselpwd 5197 . 2 (𝜑𝐵 ∈ 𝒫 𝐴)
93, 6, 8rspcdva 3544 1 (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112  ∀wral 3071   ∩ cin 3858   ⊆ wss 3859  𝒫 cpw 4495  ∪ cuni 4799  ∩ cint 4839  Moorecmoore 34791 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rab 3080  df-v 3412  df-dif 3862  df-in 3866  df-ss 3876  df-nul 4227  df-pw 4497  df-uni 4800  df-int 4840  df-bj-moore 34792 This theorem is referenced by:  bj-ismoored2  34796
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