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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpglem1 | Structured version Visualization version GIF version |
Description: Lemma for elpg 47245. (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 4568 | . . . . 5 ⊢ ((1st ‘𝐴) ∈ 𝒫 𝑥 → (1st ‘𝐴) ⊆ 𝑥) | |
2 | 1 | adantl 483 | . . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (1st ‘𝐴) ∈ 𝒫 𝑥) → (1st ‘𝐴) ⊆ 𝑥) |
3 | simpl 484 | . . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (1st ‘𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg) | |
4 | 2, 3 | sstrd 3955 | . . 3 ⊢ ((𝑥 ⊆ Pg ∧ (1st ‘𝐴) ∈ 𝒫 𝑥) → (1st ‘𝐴) ⊆ Pg) |
5 | elpwi 4568 | . . . . 5 ⊢ ((2nd ‘𝐴) ∈ 𝒫 𝑥 → (2nd ‘𝐴) ⊆ 𝑥) | |
6 | 5 | adantl 483 | . . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥) → (2nd ‘𝐴) ⊆ 𝑥) |
7 | simpl 484 | . . . 4 ⊢ ((𝑥 ⊆ Pg ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥) → 𝑥 ⊆ Pg) | |
8 | 6, 7 | sstrd 3955 | . . 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 1934 | 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 397 ∃wex 1782 ∈ wcel 2107 ⊆ wss 3911 𝒫 cpw 4561 ‘cfv 6497 1st c1st 7920 2nd c2nd 7921 Pgcpg 47240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-in 3918 df-ss 3928 df-pw 4563 |
This theorem is referenced by: elpg 47245 |
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