elpglem1 - Mathbox for Emmett Weisz

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Theorem elpglem1 49555

Description: Lemma for elpg 49558. (Contributed by Emmett Weisz, 28-Aug-2021.)

**Assertion**
Ref	Expression
elpglem1	⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1^st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥)) → ((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg))

Distinct variable group: 𝑥,𝐴

**Proof of Theorem elpglem1**
Step	Hyp	Ref	Expression
1		elpwi 4587	. . . . 5 ⊢ ((1^st ‘𝐴) ∈ 𝒫 𝑥 → (1^st ‘𝐴) ⊆ 𝑥)
2	1	adantl 481	. . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (1^st ‘𝐴) ∈ 𝒫 𝑥) → (1^st ‘𝐴) ⊆ 𝑥)
3		simpl 482	. . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (1^st ‘𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg)
4	2, 3	sstrd 3974	. . 3 ⊢ ((𝑥 ⊆ Pg ∧ (1^st ‘𝐴) ∈ 𝒫 𝑥) → (1^st ‘𝐴) ⊆ Pg)
5		elpwi 4587	. . . . 5 ⊢ ((2^nd ‘𝐴) ∈ 𝒫 𝑥 → (2^nd ‘𝐴) ⊆ 𝑥)
6	5	adantl 481	. . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥) → (2^nd ‘𝐴) ⊆ 𝑥)
7		simpl 482	. . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg)
8	6, 7	sstrd 3974	. . 3 ⊢ ((𝑥 ⊆ Pg ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥) → (2^nd ‘𝐴) ⊆ Pg)
9	4, 8	anim12dan 619	. 2 ⊢ ((𝑥 ⊆ Pg ∧ ((1^st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥)) → ((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg))
10	9	exlimiv 1930	1 ⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1^st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥)) → ((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∃wex 1779 ∈ wcel 2109 ⊆ wss 3931 𝒫 cpw 4580 ‘cfv 6536 1^st c1st 7991 2^nd c2nd 7992 Pgcpg 49553

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2708

This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2066 df-clab 2715 df-cleq 2728 df-clel 2810 df-ss 3948 df-pw 4582

This theorem is referenced by: elpg 49558