Theorem elpglem2 48476

Description: Lemma for elpg 48478. (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 6916	. . . . 5 ⊢ (1st ‘𝐴) ∈ V
2		fvex 6916	. . . . 5 ⊢ (2nd ‘𝐴) ∈ V
3	1, 2	unex 7756	. . . 4 ⊢ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ∈ V
4	3	isseti 3479	. . 3 ⊢ ∃𝑥 𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴))
5		sseq1 4005	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ⊆ Pg))
6		unss 4185	. . . . . 6 ⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) ↔ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ⊆ Pg)
7	5, 6	bitr4di 288	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)))
8	7	biimprd 247	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → 𝑥 ⊆ Pg))
9		ssun1 4173	. . . . . . 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 4033	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (1st ‘𝐴) ⊆ 𝑥)
12		vex 3466	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	elpw2 5354	. . . . . 6 ⊢ ((1st ‘𝐴) ∈ 𝒫 𝑥 ↔ (1st ‘𝐴) ⊆ 𝑥)
14	11, 13	sylibr 233	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (1st ‘𝐴) ∈ 𝒫 𝑥)
15		ssun2 4174	. . . . . . 7 ⊢ (2nd ‘𝐴) ⊆ ((1st ‘𝐴) ∪ (2nd ‘𝐴))
16	15, 10	sseqtrrid 4033	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (2nd ‘𝐴) ⊆ 𝑥)
17	12	elpw2 5354	. . . . . 6 ⊢ ((2nd ‘𝐴) ∈ 𝒫 𝑥 ↔ (2nd ‘𝐴) ⊆ 𝑥)
18	16, 17	sylibr 233	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (2nd ‘𝐴) ∈ 𝒫 𝑥)
19	14, 18	jca 510	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))
20	8, 19	jctird 525	. . 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 1832	. 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 394   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ∪ cun 3945   ⊆ wss 3947  𝒫 cpw 4607  ‘cfv 6556  1^st c1st 8003  2^nd c2nd 8004  Pgcpg 48473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-pw 4609  df-sn 4634  df-pr 4636  df-uni 4916  df-iota 6508  df-fv 6564

This theorem is referenced by:  elpg  48478