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Theorem elpglem2 47747
Description: Lemma for elpg 47749. (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 6904 . . . . 5 (1st𝐴) ∈ V
2 fvex 6904 . . . . 5 (2nd𝐴) ∈ V
31, 2unex 7732 . . . 4 ((1st𝐴) ∪ (2nd𝐴)) ∈ V
43isseti 3489 . . 3 𝑥 𝑥 = ((1st𝐴) ∪ (2nd𝐴))
5 sseq1 4007 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (𝑥 ⊆ Pg ↔ ((1st𝐴) ∪ (2nd𝐴)) ⊆ Pg))
6 unss 4184 . . . . . 6 (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) ↔ ((1st𝐴) ∪ (2nd𝐴)) ⊆ Pg)
75, 6bitr4di 288 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (𝑥 ⊆ Pg ↔ ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg)))
87biimprd 247 . . . 4 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → 𝑥 ⊆ Pg))
9 ssun1 4172 . . . . . . 7 (1st𝐴) ⊆ ((1st𝐴) ∪ (2nd𝐴))
10 id 22 . . . . . . 7 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → 𝑥 = ((1st𝐴) ∪ (2nd𝐴)))
119, 10sseqtrrid 4035 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (1st𝐴) ⊆ 𝑥)
12 vex 3478 . . . . . . 7 𝑥 ∈ V
1312elpw2 5345 . . . . . 6 ((1st𝐴) ∈ 𝒫 𝑥 ↔ (1st𝐴) ⊆ 𝑥)
1411, 13sylibr 233 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (1st𝐴) ∈ 𝒫 𝑥)
15 ssun2 4173 . . . . . . 7 (2nd𝐴) ⊆ ((1st𝐴) ∪ (2nd𝐴))
1615, 10sseqtrrid 4035 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (2nd𝐴) ⊆ 𝑥)
1712elpw2 5345 . . . . . 6 ((2nd𝐴) ∈ 𝒫 𝑥 ↔ (2nd𝐴) ⊆ 𝑥)
1816, 17sylibr 233 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (2nd𝐴) ∈ 𝒫 𝑥)
1914, 18jca 512 . . . 4 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))
208, 19jctird 527 . . 3 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
214, 20eximii 1839 . 2 𝑥(((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
222119.37iv 1952 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 396   = wceq 1541  wex 1781  wcel 2106  cun 3946  wss 3948  𝒫 cpw 4602  cfv 6543  1st c1st 7972  2nd c2nd 7973  Pgcpg 47744
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  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-pw 4604  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-fv 6551
This theorem is referenced by:  elpg  47749
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