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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpglem1 | Structured version Visualization version GIF version |
Description: Lemma for elpg 44810. (Contributed by Emmett Weisz, 28-Aug-2021.) |
Ref | Expression |
---|---|
elpglem1 | ⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)) → ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwi 4550 | . . . . 5 ⊢ ((1st ‘𝐴) ∈ 𝒫 𝑥 → (1st ‘𝐴) ⊆ 𝑥) | |
2 | 1 | adantl 484 | . . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (1st ‘𝐴) ∈ 𝒫 𝑥) → (1st ‘𝐴) ⊆ 𝑥) |
3 | simpl 485 | . . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (1st ‘𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg) | |
4 | 2, 3 | sstrd 3976 | . . 3 ⊢ ((𝑥 ⊆ Pg ∧ (1st ‘𝐴) ∈ 𝒫 𝑥) → (1st ‘𝐴) ⊆ Pg) |
5 | elpwi 4550 | . . . . 5 ⊢ ((2nd ‘𝐴) ∈ 𝒫 𝑥 → (2nd ‘𝐴) ⊆ 𝑥) | |
6 | 5 | adantl 484 | . . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥) → (2nd ‘𝐴) ⊆ 𝑥) |
7 | simpl 485 | . . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg) | |
8 | 6, 7 | sstrd 3976 | . . 3 ⊢ ((𝑥 ⊆ Pg ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥) → (2nd ‘𝐴) ⊆ Pg) |
9 | 4, 8 | anim12dan 620 | . 2 ⊢ ((𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)) → ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)) |
10 | 9 | exlimiv 1927 | 1 ⊢ (∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)) → ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∃wex 1776 ∈ wcel 2110 ⊆ wss 3935 𝒫 cpw 4538 ‘cfv 6349 1st c1st 7681 2nd 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 |
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-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-in 3942 df-ss 3951 df-pw 4540 |
This theorem is referenced by: elpg 44810 |
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