elpglem2 - Mathbox for Emmett Weisz

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Theorem elpglem2 51211

Description: Lemma for elpg 51213. (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 6929	. . . . 5 ⊢ (1^st ‘𝐴) ∈ V
2		fvex 6929	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
3	1, 2	unex 7793	. . . 4 ⊢ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) ∈ V
4	3	isseti 3472	. . 3 ⊢ ∃𝑥 𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴))
5		sseq1 3938	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) ⊆ Pg))
6		unss 4191	. . . . . 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 4179	. . . . . . 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 3956	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (1^st ‘𝐴) ⊆ 𝑥)
12		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	elpw2 5303	. . . . . 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 4180	. . . . . . 7 ⊢ (2^nd ‘𝐴) ⊆ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴))
16	15, 10	sseqtrrid 3956	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (2^nd ‘𝐴) ⊆ 𝑥)
17	12	elpw2 5303	. . . . . 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 521	. . . 4 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → ((1^st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥))
20	8, 19	jctird 536	. . 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 1870	. 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 1983	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2147 ⊆ wss 3900 ∪ cun 4152 𝒫 cpw 4560 ‘cfv 6566 1^st c1st 8033 2^nd c2nd 8034 Pgcpg 51208

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2102 df-clab 2742 df-cleq 2755 df-clel 2839 df-ne 2960 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-pw 4562 df-sn 4589 df-pr 4591 df-uni 4873 df-iota 6520 df-fv 6574

This theorem is used by: elpg 51213

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