elpglem2 - Mathbox for Emmett Weisz

	Mathbox for Emmett Weisz		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > elpglem2		Structured version Visualization version GIF version

Theorem elpglem2 50368

Description: Lemma for elpg 50370. (Contributed by Emmett Weisz, 28-Aug-2021.)

**Assertion**
Ref	Expression
elpglem2	⊢ (((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1^st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥)))

Distinct variable group: 𝑥,𝐴

**Proof of Theorem elpglem2**
Step	Hyp	Ref	Expression
1		fvex 6892	. . . . 5 ⊢ (1^st ‘𝐴) ∈ V
2		fvex 6892	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
3	1, 2	unex 7739	. . . 4 ⊢ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) ∈ V
4	3	isseti 3481	. . 3 ⊢ ∃𝑥 𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴))
5		sseq1 3970	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) ⊆ Pg))
6		unss 4151	. . . . . 6 ⊢ (((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg) ↔ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) ⊆ Pg)
7	5, 6	bitr4di 292	. . . . 5 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg)))
8	7	biimprd 251	. . . 4 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg) → 𝑥 ⊆ Pg))
9		ssun1 4139	. . . . . . 7 ⊢ (1^st ‘𝐴) ⊆ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴))
10		id 23	. . . . . . 7 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → 𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)))
11	9, 10	sseqtrrid 3988	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (1^st ‘𝐴) ⊆ 𝑥)
12		vex 3467	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	elpw2 5302	. . . . . 6 ⊢ ((1^st ‘𝐴) ∈ 𝒫 𝑥 ↔ (1^st ‘𝐴) ⊆ 𝑥)
14	11, 13	sylibr 237	. . . . 5 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (1^st ‘𝐴) ∈ 𝒫 𝑥)
15		ssun2 4140	. . . . . . 7 ⊢ (2^nd ‘𝐴) ⊆ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴))
16	15, 10	sseqtrrid 3988	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (2^nd ‘𝐴) ⊆ 𝑥)
17	12	elpw2 5302	. . . . . 6 ⊢ ((2^nd ‘𝐴) ∈ 𝒫 𝑥 ↔ (2^nd ‘𝐴) ⊆ 𝑥)
18	16, 17	sylibr 237	. . . . 5 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (2^nd ‘𝐴) ∈ 𝒫 𝑥)
19	14, 18	jca 520	. . . 4 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → ((1^st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥))
20	8, 19	jctird 535	. . 3 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1^st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥))))
21	4, 20	eximii 1864	. 2 ⊢ ∃𝑥(((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1^st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥)))
22	21	19.37iv 1975	1 ⊢ (((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1^st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∃wex 1806 ∈ wcel 2149 ∪ cun 3911 ⊆ wss 3913 𝒫 cpw 4564 ‘cfv 6533 1^st c1st 7980 2^nd c2nd 7981 Pgcpg 50365

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-ext 2741 ax-sep 5258 ax-nul 5268 ax-pr 5402 ax-un 7730

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-sb 2098 df-clab 2748 df-cleq 2761 df-clel 2844 df-ne 2965 df-rab 3424 df-v 3465 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-pw 4566 df-sn 4592 df-pr 4594 df-uni 4874 df-iota 6489 df-fv 6541

This theorem is referenced by: elpg 50370

W3C validator