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Theorem bj-ismoored 33387
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 bj-ismoored.2 . 2 (𝜑𝐵𝐴)
2 bj-ismoored.1 . . 3 (𝜑𝐴Moore)
3 bj-ismoorec 33385 . . 3 (𝐴Moore ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
42, 3sylib 208 . 2 (𝜑 → (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
5 elpw2g 4955 . . . . 5 (𝐴 ∈ V → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
65biimparc 465 . . . 4 ((𝐵𝐴𝐴 ∈ V) → 𝐵 ∈ 𝒫 𝐴)
7 inteq 4612 . . . . . . 7 (𝑥 = 𝐵 𝑥 = 𝐵)
87ineq2d 3963 . . . . . 6 (𝑥 = 𝐵 → ( 𝐴 𝑥) = ( 𝐴 𝐵))
98eleq1d 2834 . . . . 5 (𝑥 = 𝐵 → (( 𝐴 𝑥) ∈ 𝐴 ↔ ( 𝐴 𝐵) ∈ 𝐴))
109rspcv 3454 . . . 4 (𝐵 ∈ 𝒫 𝐴 → (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 → ( 𝐴 𝐵) ∈ 𝐴))
116, 10syl 17 . . 3 ((𝐵𝐴𝐴 ∈ V) → (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 → ( 𝐴 𝐵) ∈ 𝐴))
1211expimpd 441 . 2 (𝐵𝐴 → ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴) → ( 𝐴 𝐵) ∈ 𝐴))
131, 4, 12sylc 65 1 (𝜑 → ( 𝐴 𝐵) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  Vcvv 3349  cin 3720  wss 3721  𝒫 cpw 4295   cuni 4572   cint 4609  Moorecmoore 33382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-v 3351  df-in 3728  df-ss 3735  df-pw 4297  df-uni 4573  df-int 4610  df-bj-moore 33383
This theorem is referenced by:  bj-ismoored2  33388
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