Theorem elpglem2 46088

Description: Lemma for elpg 46090. (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 6730	. . . . 5 ⊢ (1st ‘𝐴) ∈ V
2		fvex 6730	. . . . 5 ⊢ (2nd ‘𝐴) ∈ V
3	1, 2	unex 7531	. . . 4 ⊢ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ∈ V
4	3	isseti 3423	. . 3 ⊢ ∃𝑥 𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴))
5		sseq1 3926	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ⊆ Pg))
6		unss 4098	. . . . . 6 ⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) ↔ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ⊆ Pg)
7	5, 6	bitr4di 292	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)))
8	7	biimprd 251	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → 𝑥 ⊆ Pg))
9		ssun1 4086	. . . . . . 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 3954	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (1st ‘𝐴) ⊆ 𝑥)
12		vex 3412	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	elpw2 5238	. . . . . 6 ⊢ ((1st ‘𝐴) ∈ 𝒫 𝑥 ↔ (1st ‘𝐴) ⊆ 𝑥)
14	11, 13	sylibr 237	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (1st ‘𝐴) ∈ 𝒫 𝑥)
15		ssun2 4087	. . . . . . 7 ⊢ (2nd ‘𝐴) ⊆ ((1st ‘𝐴) ∪ (2nd ‘𝐴))
16	15, 10	sseqtrrid 3954	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (2nd ‘𝐴) ⊆ 𝑥)
17	12	elpw2 5238	. . . . . 6 ⊢ ((2nd ‘𝐴) ∈ 𝒫 𝑥 ↔ (2nd ‘𝐴) ⊆ 𝑥)
18	16, 17	sylibr 237	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (2nd ‘𝐴) ∈ 𝒫 𝑥)
19	14, 18	jca 515	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))
20	8, 19	jctird 530	. . 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 1844	. 2 ⊢ ∃𝑥(((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))
22	21	19.37iv 1957	1 ⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2110   ∪ cun 3864   ⊆ wss 3866  𝒫 cpw 4513  ‘cfv 6380  1^st c1st 7759  2^nd c2nd 7760  Pgcpg 46085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-pw 4515  df-sn 4542  df-pr 4544  df-uni 4820  df-iota 6338  df-fv 6388

This theorem is referenced by:  elpg  46090