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Theorem elpglem1 47242
Description: Lemma for elpg 47245. (Contributed by Emmett Weisz, 28-Aug-2021.)
Assertion
Ref Expression
elpglem1 (∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
Distinct variable group:   𝑥,𝐴

Proof of Theorem elpglem1
StepHypRef Expression
1 elpwi 4568 . . . . 5 ((1st𝐴) ∈ 𝒫 𝑥 → (1st𝐴) ⊆ 𝑥)
21adantl 483 . . . 4 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → (1st𝐴) ⊆ 𝑥)
3 simpl 484 . . . 4 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg)
42, 3sstrd 3955 . . 3 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → (1st𝐴) ⊆ Pg)
5 elpwi 4568 . . . . 5 ((2nd𝐴) ∈ 𝒫 𝑥 → (2nd𝐴) ⊆ 𝑥)
65adantl 483 . . . 4 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → (2nd𝐴) ⊆ 𝑥)
7 simpl 484 . . . 4 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg)
86, 7sstrd 3955 . . 3 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → (2nd𝐴) ⊆ Pg)
94, 8anim12dan 620 . 2 ((𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
109exlimiv 1934 1 (∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wex 1782  wcel 2107  wss 3911  𝒫 cpw 4561  cfv 6497  1st c1st 7920  2nd c2nd 7921  Pgcpg 47240
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-in 3918  df-ss 3928  df-pw 4563
This theorem is referenced by:  elpg  47245
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