Theorem elpglem2 44808

Description: Lemma for elpg 44810. (Contributed by Emmett Weisz, 28-Aug-2021.)

Assertion

Ref	Expression
elpglem2	⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))

Distinct variable group:   𝑥,𝐴

Proof of Theorem elpglem2

Step	Hyp	Ref	Expression
1		fvex 6677	. . . . 5 ⊢ (1st ‘𝐴) ∈ V
2		fvex 6677	. . . . 5 ⊢ (2nd ‘𝐴) ∈ V
3	1, 2	unex 7463	. . . 4 ⊢ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ∈ V
4	3	isseti 3508	. . 3 ⊢ ∃𝑥 𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴))
5		sseq1 3991	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ⊆ Pg))
6		unss 4159	. . . . . 6 ⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) ↔ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ⊆ Pg)
7	5, 6	syl6bbr 291	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)))
8	7	biimprd 250	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → 𝑥 ⊆ Pg))
9		ssun1 4147	. . . . . . 7 ⊢ (1st ‘𝐴) ⊆ ((1st ‘𝐴) ∪ (2nd ‘𝐴))
10		id 22	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → 𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)))
11	9, 10	sseqtrrid 4019	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (1st ‘𝐴) ⊆ 𝑥)
12		vex 3497	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	elpw2 5240	. . . . . 6 ⊢ ((1st ‘𝐴) ∈ 𝒫 𝑥 ↔ (1st ‘𝐴) ⊆ 𝑥)
14	11, 13	sylibr 236	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (1st ‘𝐴) ∈ 𝒫 𝑥)
15		ssun2 4148	. . . . . . 7 ⊢ (2nd ‘𝐴) ⊆ ((1st ‘𝐴) ∪ (2nd ‘𝐴))
16	15, 10	sseqtrrid 4019	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (2nd ‘𝐴) ⊆ 𝑥)
17	12	elpw2 5240	. . . . . 6 ⊢ ((2nd ‘𝐴) ∈ 𝒫 𝑥 ↔ (2nd ‘𝐴) ⊆ 𝑥)
18	16, 17	sylibr 236	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (2nd ‘𝐴) ∈ 𝒫 𝑥)
19	14, 18	jca 514	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))
20	8, 19	jctird 529	. . 3 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))))
21	4, 20	eximii 1833	. 2 ⊢ ∃𝑥(((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))
22	21	19.37iv 1945	1 ⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ∪ cun 3933   ⊆ wss 3935  𝒫 cpw 4538  ‘cfv 6349  1^st c1st 7681  2^nd c2nd 7682  Pgcpg 44805

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-sn 4561  df-pr 4563  df-uni 4832  df-iota 6308  df-fv 6357

This theorem is referenced by:  elpg  44810