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Theorem elpglem1 47709
Description: Lemma for elpg 47712. (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 4608 . . . . 5 ((1st𝐴) ∈ 𝒫 𝑥 → (1st𝐴) ⊆ 𝑥)
21adantl 482 . . . 4 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → (1st𝐴) ⊆ 𝑥)
3 simpl 483 . . . 4 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg)
42, 3sstrd 3991 . . 3 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → (1st𝐴) ⊆ Pg)
5 elpwi 4608 . . . . 5 ((2nd𝐴) ∈ 𝒫 𝑥 → (2nd𝐴) ⊆ 𝑥)
65adantl 482 . . . 4 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → (2nd𝐴) ⊆ 𝑥)
7 simpl 483 . . . 4 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg)
86, 7sstrd 3991 . . 3 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → (2nd𝐴) ⊆ Pg)
94, 8anim12dan 619 . 2 ((𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
109exlimiv 1933 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 396  wex 1781  wcel 2106  wss 3947  𝒫 cpw 4601  cfv 6540  1st c1st 7969  2nd c2nd 7970  Pgcpg 47707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-ss 3964  df-pw 4603
This theorem is referenced by:  elpg  47712
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