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Theorem elpglem2 48943
Description: Lemma for elpg 48945. (Contributed by Emmett Weisz, 28-Aug-2021.)
Assertion
Ref Expression
elpglem2 (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem elpglem2
StepHypRef Expression
1 fvex 6920 . . . . 5 (1st𝐴) ∈ V
2 fvex 6920 . . . . 5 (2nd𝐴) ∈ V
31, 2unex 7763 . . . 4 ((1st𝐴) ∪ (2nd𝐴)) ∈ V
43isseti 3496 . . 3 𝑥 𝑥 = ((1st𝐴) ∪ (2nd𝐴))
5 sseq1 4021 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (𝑥 ⊆ Pg ↔ ((1st𝐴) ∪ (2nd𝐴)) ⊆ Pg))
6 unss 4200 . . . . . 6 (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) ↔ ((1st𝐴) ∪ (2nd𝐴)) ⊆ Pg)
75, 6bitr4di 289 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (𝑥 ⊆ Pg ↔ ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg)))
87biimprd 248 . . . 4 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → 𝑥 ⊆ Pg))
9 ssun1 4188 . . . . . . 7 (1st𝐴) ⊆ ((1st𝐴) ∪ (2nd𝐴))
10 id 22 . . . . . . 7 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → 𝑥 = ((1st𝐴) ∪ (2nd𝐴)))
119, 10sseqtrrid 4049 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (1st𝐴) ⊆ 𝑥)
12 vex 3482 . . . . . . 7 𝑥 ∈ V
1312elpw2 5340 . . . . . 6 ((1st𝐴) ∈ 𝒫 𝑥 ↔ (1st𝐴) ⊆ 𝑥)
1411, 13sylibr 234 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (1st𝐴) ∈ 𝒫 𝑥)
15 ssun2 4189 . . . . . . 7 (2nd𝐴) ⊆ ((1st𝐴) ∪ (2nd𝐴))
1615, 10sseqtrrid 4049 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (2nd𝐴) ⊆ 𝑥)
1712elpw2 5340 . . . . . 6 ((2nd𝐴) ∈ 𝒫 𝑥 ↔ (2nd𝐴) ⊆ 𝑥)
1816, 17sylibr 234 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (2nd𝐴) ∈ 𝒫 𝑥)
1914, 18jca 511 . . . 4 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))
208, 19jctird 526 . . 3 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
214, 20eximii 1834 . 2 𝑥(((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
222119.37iv 1946 1 (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  cun 3961  wss 3963  𝒫 cpw 4605  cfv 6563  1st c1st 8011  2nd c2nd 8012  Pgcpg 48940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516  df-fv 6571
This theorem is referenced by:  elpg  48945
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