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Theorem elpglem1 42225
Description: Lemma for elpg 42228. (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 4166 . . . . 5 ((1st𝐴) ∈ 𝒫 𝑥 → (1st𝐴) ⊆ 𝑥)
21adantl 482 . . . 4 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → (1st𝐴) ⊆ 𝑥)
3 simpl 473 . . . 4 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg)
42, 3sstrd 3611 . . 3 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → (1st𝐴) ⊆ Pg)
5 elpwi 4166 . . . . 5 ((2nd𝐴) ∈ 𝒫 𝑥 → (2nd𝐴) ⊆ 𝑥)
65adantl 482 . . . 4 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → (2nd𝐴) ⊆ 𝑥)
7 simpl 473 . . . 4 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg)
86, 7sstrd 3611 . . 3 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → (2nd𝐴) ⊆ Pg)
94, 8anim12dan 882 . 2 ((𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
109exlimiv 1857 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 384  wex 1703  wcel 1989  wss 3572  𝒫 cpw 4156  cfv 5886  1st c1st 7163  2nd c2nd 7164  Pgcpg 42223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-v 3200  df-in 3579  df-ss 3586  df-pw 4158
This theorem is referenced by:  elpg  42228
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