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Theorem elpglem2 47243
Description: Lemma for elpg 47245. (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 6856 . . . . 5 (1st𝐴) ∈ V
2 fvex 6856 . . . . 5 (2nd𝐴) ∈ V
31, 2unex 7681 . . . 4 ((1st𝐴) ∪ (2nd𝐴)) ∈ V
43isseti 3459 . . 3 𝑥 𝑥 = ((1st𝐴) ∪ (2nd𝐴))
5 sseq1 3970 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (𝑥 ⊆ Pg ↔ ((1st𝐴) ∪ (2nd𝐴)) ⊆ Pg))
6 unss 4145 . . . . . 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 4133 . . . . . . 7 (1st𝐴) ⊆ ((1st𝐴) ∪ (2nd𝐴))
10 id 22 . . . . . . 7 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → 𝑥 = ((1st𝐴) ∪ (2nd𝐴)))
119, 10sseqtrrid 3998 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (1st𝐴) ⊆ 𝑥)
12 vex 3448 . . . . . . 7 𝑥 ∈ V
1312elpw2 5303 . . . . . 6 ((1st𝐴) ∈ 𝒫 𝑥 ↔ (1st𝐴) ⊆ 𝑥)
1411, 13sylibr 233 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (1st𝐴) ∈ 𝒫 𝑥)
15 ssun2 4134 . . . . . . 7 (2nd𝐴) ⊆ ((1st𝐴) ∪ (2nd𝐴))
1615, 10sseqtrrid 3998 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (2nd𝐴) ⊆ 𝑥)
1712elpw2 5303 . . . . . 6 ((2nd𝐴) ∈ 𝒫 𝑥 ↔ (2nd𝐴) ⊆ 𝑥)
1816, 17sylibr 233 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (2nd𝐴) ∈ 𝒫 𝑥)
1914, 18jca 513 . . . 4 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))
208, 19jctird 528 . . 3 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
214, 20eximii 1840 . 2 𝑥(((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
222119.37iv 1953 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 397   = wceq 1542  wex 1782  wcel 2107  cun 3909  wss 3911  𝒫 cpw 4561  cfv 6497  1st c1st 7920  2nd c2nd 7921  Pgcpg 47240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-pw 4563  df-sn 4588  df-pr 4590  df-uni 4867  df-iota 6449  df-fv 6505
This theorem is referenced by:  elpg  47245
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