elpglem2 - Mathbox for Emmett Weisz

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Theorem elpglem2 50987

Description: Lemma for elpg 50989. (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 6901	. . . . 5 ⊢ (1^st ‘𝐴) ∈ V
2		fvex 6901	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
3	1, 2	unex 7763	. . . 4 ⊢ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) ∈ V
4	3	isseti 3468	. . 3 ⊢ ∃𝑥 𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴))
5		sseq1 3934	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) ⊆ Pg))
6		unss 4187	. . . . . 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 4175	. . . . . . 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 3952	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (1^st ‘𝐴) ⊆ 𝑥)
12		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	elpw2 5295	. . . . . 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 4176	. . . . . . 7 ⊢ (2^nd ‘𝐴) ⊆ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴))
16	15, 10	sseqtrrid 3952	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (2^nd ‘𝐴) ⊆ 𝑥)
17	12	elpw2 5295	. . . . . 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 1869	. 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 1980	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 1569 ∃wex 1811 ∈ wcel 2144 ⊆ wss 3896 ∪ cun 4148 𝒫 cpw 4556 ‘cfv 6541 1^st c1st 8001 2^nd c2nd 8002 Pgcpg 50984

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-un 7753

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-sb 2099 df-clab 2739 df-cleq 2752 df-clel 2835 df-ne 2956 df-rab 3413 df-v 3452 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-pw 4558 df-sn 4585 df-pr 4587 df-uni 4867 df-iota 6496 df-fv 6549

This theorem is used by: elpg 50989

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