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Theorem elpglem2 41748
Description: Lemma for elpg 41750. (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 6158 . . . . 5 (1st𝐴) ∈ V
2 fvex 6158 . . . . 5 (2nd𝐴) ∈ V
31, 2unex 6909 . . . 4 ((1st𝐴) ∪ (2nd𝐴)) ∈ V
43isseti 3195 . . 3 𝑥 𝑥 = ((1st𝐴) ∪ (2nd𝐴))
5 sseq1 3605 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (𝑥 ⊆ Pg ↔ ((1st𝐴) ∪ (2nd𝐴)) ⊆ Pg))
6 unss 3765 . . . . . 6 (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) ↔ ((1st𝐴) ∪ (2nd𝐴)) ⊆ Pg)
75, 6syl6bbr 278 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (𝑥 ⊆ Pg ↔ ((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg)))
87biimprd 238 . . . 4 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → 𝑥 ⊆ Pg))
9 ssun1 3754 . . . . . . 7 (1st𝐴) ⊆ ((1st𝐴) ∪ (2nd𝐴))
10 id 22 . . . . . . 7 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → 𝑥 = ((1st𝐴) ∪ (2nd𝐴)))
119, 10syl5sseqr 3633 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (1st𝐴) ⊆ 𝑥)
12 vex 3189 . . . . . . 7 𝑥 ∈ V
1312elpw2 4788 . . . . . 6 ((1st𝐴) ∈ 𝒫 𝑥 ↔ (1st𝐴) ⊆ 𝑥)
1411, 13sylibr 224 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (1st𝐴) ∈ 𝒫 𝑥)
15 ssun2 3755 . . . . . . 7 (2nd𝐴) ⊆ ((1st𝐴) ∪ (2nd𝐴))
1615, 10syl5sseqr 3633 . . . . . 6 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (2nd𝐴) ⊆ 𝑥)
1712elpw2 4788 . . . . . 6 ((2nd𝐴) ∈ 𝒫 𝑥 ↔ (2nd𝐴) ⊆ 𝑥)
1816, 17sylibr 224 . . . . 5 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (2nd𝐴) ∈ 𝒫 𝑥)
1914, 18jca 554 . . . 4 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))
208, 19jctird 566 . . 3 (𝑥 = ((1st𝐴) ∪ (2nd𝐴)) → (((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥))))
214, 20eximii 1761 . 2 𝑥(((1st𝐴) ⊆ Pg ∧ (2nd𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st𝐴) ∈ 𝒫 𝑥 ∧ (2nd𝐴) ∈ 𝒫 𝑥)))
222119.37iv 1908 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 384   = wceq 1480  wex 1701  wcel 1987  cun 3553  wss 3555  𝒫 cpw 4130  cfv 5847  1st c1st 7111  2nd c2nd 7112  Pgcpg 41745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-pw 4132  df-sn 4149  df-pr 4151  df-uni 4403  df-iota 5810  df-fv 5855
This theorem is referenced by:  elpg  41750
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