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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpglem1 | Structured version Visualization version GIF version |
Description: Lemma for elpg 48015. (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 4604 | . . . . 5 ⊢ ((1st ‘𝐴) ∈ 𝒫 𝑥 → (1st ‘𝐴) ⊆ 𝑥) | |
2 | 1 | adantl 481 | . . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (1st ‘𝐴) ∈ 𝒫 𝑥) → (1st ‘𝐴) ⊆ 𝑥) |
3 | simpl 482 | . . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (1st ‘𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg) | |
4 | 2, 3 | sstrd 3987 | . . 3 ⊢ ((𝑥 ⊆ Pg ∧ (1st ‘𝐴) ∈ 𝒫 𝑥) → (1st ‘𝐴) ⊆ Pg) |
5 | elpwi 4604 | . . . . 5 ⊢ ((2nd ‘𝐴) ∈ 𝒫 𝑥 → (2nd ‘𝐴) ⊆ 𝑥) | |
6 | 5 | adantl 481 | . . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥) → (2nd ‘𝐴) ⊆ 𝑥) |
7 | simpl 482 | . . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg) | |
8 | 6, 7 | sstrd 3987 | . . 3 ⊢ ((𝑥 ⊆ Pg ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥) → (2nd ‘𝐴) ⊆ Pg) |
9 | 4, 8 | anim12dan 618 | . 2 ⊢ ((𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)) → ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)) |
10 | 9 | exlimiv 1925 | 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 395 ∃wex 1773 ∈ wcel 2098 ⊆ wss 3943 𝒫 cpw 4597 ‘cfv 6536 1st c1st 7969 2nd c2nd 7970 Pgcpg 48010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-v 3470 df-in 3950 df-ss 3960 df-pw 4599 |
This theorem is referenced by: elpg 48015 |
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