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Theorem elpglem2 47844
Description: Lemma for elpg 47846. (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 6903 . . . . 5 (1st𝐴) ∈ V
2 fvex 6903 . . . . 5 (2nd𝐴) ∈ V
31, 2unex 7735 . . . 4 ((1st𝐴) ∪ (2nd𝐴)) ∈ V
43isseti 3488 . . 3 𝑥 𝑥 = ((1st𝐴) ∪ (2nd𝐴))
5 sseq1 4006 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (𝑥 ⊆ Pg ↔ ((1st𝐴) ∪ (2nd𝐴)) ⊆ Pg))
6 unss 4183 . . . . . 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 4171 . . . . . . 7 (1st𝐴) ⊆ ((1st𝐴) ∪ (2nd𝐴))
10 id 22 . . . . . . 7 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → 𝑥 = ((1st𝐴) ∪ (2nd𝐴)))
119, 10sseqtrrid 4034 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (1st𝐴) ⊆ 𝑥)
12 vex 3476 . . . . . . 7 𝑥 ∈ V
1312elpw2 5344 . . . . . 6 ((1st𝐴) ∈ 𝒫 𝑥 ↔ (1st𝐴) ⊆ 𝑥)
1411, 13sylibr 233 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (1st𝐴) ∈ 𝒫 𝑥)
15 ssun2 4172 . . . . . . 7 (2nd𝐴) ⊆ ((1st𝐴) ∪ (2nd𝐴))
1615, 10sseqtrrid 4034 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (2nd𝐴) ⊆ 𝑥)
1712elpw2 5344 . . . . . 6 ((2nd𝐴) ∈ 𝒫 𝑥 ↔ (2nd𝐴) ⊆ 𝑥)
1816, 17sylibr 233 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (2nd𝐴) ∈ 𝒫 𝑥)
1914, 18jca 510 . . . 4 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))
208, 19jctird 525 . . 3 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
214, 20eximii 1837 . 2 𝑥(((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
222119.37iv 1950 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 394   = wceq 1539  wex 1779  wcel 2104  cun 3945  wss 3947  𝒫 cpw 4601  cfv 6542  1st c1st 7975  2nd c2nd 7976  Pgcpg 47841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-pw 4603  df-sn 4628  df-pr 4630  df-uni 4908  df-iota 6494  df-fv 6550
This theorem is referenced by:  elpg  47846
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