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Theorem elpglem1 48012
Description: Lemma for elpg 48015. (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 4604 . . . . 5 ((1st𝐴) ∈ 𝒫 𝑥 → (1st𝐴) ⊆ 𝑥)
21adantl 481 . . . 4 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → (1st𝐴) ⊆ 𝑥)
3 simpl 482 . . . 4 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg)
42, 3sstrd 3987 . . 3 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → (1st𝐴) ⊆ Pg)
5 elpwi 4604 . . . . 5 ((2nd𝐴) ∈ 𝒫 𝑥 → (2nd𝐴) ⊆ 𝑥)
65adantl 481 . . . 4 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → (2nd𝐴) ⊆ 𝑥)
7 simpl 482 . . . 4 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg)
86, 7sstrd 3987 . . 3 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → (2nd𝐴) ⊆ Pg)
94, 8anim12dan 618 . 2 ((𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
109exlimiv 1925 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 395  wex 1773  wcel 2098  wss 3943  𝒫 cpw 4597  cfv 6536  1st c1st 7969  2nd c2nd 7970  Pgcpg 48010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599
This theorem is referenced by:  elpg  48015
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