elpglem2 - Mathbox for Emmett Weisz

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Theorem elpglem2 50473

Description: Lemma for elpg 50475. (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 6896	. . . . 5 ⊢ (1^st ‘𝐴) ∈ V
2		fvex 6896	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
3	1, 2	unex 7744	. . . 4 ⊢ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) ∈ V
4	3	isseti 3473	. . 3 ⊢ ∃𝑥 𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴))
5		sseq1 3963	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) ⊆ Pg))
6		unss 4144	. . . . . 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 4132	. . . . . . 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 3981	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (1^st ‘𝐴) ⊆ 𝑥)
12		vex 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	elpw2 5306	. . . . . 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 4133	. . . . . . 7 ⊢ (2^nd ‘𝐴) ⊆ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴))
16	15, 10	sseqtrrid 3981	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (2^nd ‘𝐴) ⊆ 𝑥)
17	12	elpw2 5306	. . . . . 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 1867	. 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 1978	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 1570 ∃wex 1809 ∈ wcel 2143 ∪ cun 3904 ⊆ wss 3906 𝒫 cpw 4563 ‘cfv 6538 1^st c1st 7985 2^nd c2nd 7986 Pgcpg 50470

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-sb 2097 df-clab 2742 df-cleq 2755 df-clel 2838 df-ne 2959 df-rab 3417 df-v 3457 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-pw 4565 df-sn 4591 df-pr 4593 df-uni 4874 df-iota 6494 df-fv 6546

This theorem is referenced by: elpg 50475

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