Mathbox for Emmett Weisz < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elpglem1 Structured version   Visualization version   GIF version

Theorem elpglem1 45254
 Description: Lemma for elpg 45257. (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 4506 . . . . 5 ((1st𝐴) ∈ 𝒫 𝑥 → (1st𝐴) ⊆ 𝑥)
21adantl 485 . . . 4 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → (1st𝐴) ⊆ 𝑥)
3 simpl 486 . . . 4 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg)
42, 3sstrd 3925 . . 3 ((𝑥 ⊆ Pg ∧ (1st𝐴) ∈ 𝒫 𝑥) → (1st𝐴) ⊆ Pg)
5 elpwi 4506 . . . . 5 ((2nd𝐴) ∈ 𝒫 𝑥 → (2nd𝐴) ⊆ 𝑥)
65adantl 485 . . . 4 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → (2nd𝐴) ⊆ 𝑥)
7 simpl 486 . . . 4 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg)
86, 7sstrd 3925 . . 3 ((𝑥 ⊆ Pg ∧ (2nd𝐴) ∈ 𝒫 𝑥) → (2nd𝐴) ⊆ Pg)
94, 8anim12dan 621 . 2 ((𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)) → ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg))
109exlimiv 1931 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 399  ∃wex 1781   ∈ wcel 2111   ⊆ wss 3881  𝒫 cpw 4497  ‘cfv 6324  1st c1st 7671  2nd c2nd 7672  Pgcpg 45252 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499 This theorem is referenced by:  elpg  45257
 Copyright terms: Public domain W3C validator