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Theorem bj-ismooredr 33370
Description: Sufficient condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.)
Hypotheses
Ref Expression
bj-ismooredr.1 (𝜑𝐴𝑉)
bj-ismooredr.2 ((𝜑𝑥𝐴) → ( 𝐴 𝑥) ∈ 𝐴)
Assertion
Ref Expression
bj-ismooredr (𝜑𝐴Moore)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-ismooredr
StepHypRef Expression
1 elpwi 4312 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
2 bj-ismooredr.2 . . . . . 6 ((𝜑𝑥𝐴) → ( 𝐴 𝑥) ∈ 𝐴)
32ex 449 . . . . 5 (𝜑 → (𝑥𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
41, 3syl5 34 . . . 4 (𝜑 → (𝑥 ∈ 𝒫 𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
54alrimiv 2004 . . 3 (𝜑 → ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
6 df-ral 3055 . . 3 (∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ( 𝐴 𝑥) ∈ 𝐴))
75, 6sylibr 224 . 2 (𝜑 → ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴)
8 bj-ismooredr.1 . . 3 (𝜑𝐴𝑉)
9 bj-ismoore 33365 . . 3 (𝐴𝑉 → (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
108, 9syl 17 . 2 (𝜑 → (𝐴Moore ↔ ∀𝑥 ∈ 𝒫 𝐴( 𝐴 𝑥) ∈ 𝐴))
117, 10mpbird 247 1 (𝜑𝐴Moore)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wcel 2139  wral 3050  cin 3714  wss 3715  𝒫 cpw 4302   cuni 4588   cint 4627  Moorecmoore 33363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-pw 4304  df-uni 4589  df-bj-moore 33364
This theorem is referenced by:  bj-discrmoore  33372
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