elpglem2 - Mathbox for Emmett Weisz

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Theorem elpglem2 50187

Description: Lemma for elpg 50189. (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 6853	. . . . 5 ⊢ (1^st ‘𝐴) ∈ V
2		fvex 6853	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
3	1, 2	unex 7698	. . . 4 ⊢ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) ∈ V
4	3	isseti 3447	. . 3 ⊢ ∃𝑥 𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴))
5		sseq1 3947	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) ⊆ Pg))
6		unss 4130	. . . . . 6 ⊢ (((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg) ↔ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) ⊆ Pg)
7	5, 6	bitr4di 289	. . . . 5 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg)))
8	7	biimprd 248	. . . 4 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (((1^st ‘𝐴) ⊆ Pg ∧ (2^nd ‘𝐴) ⊆ Pg) → 𝑥 ⊆ Pg))
9		ssun1 4118	. . . . . . 7 ⊢ (1^st ‘𝐴) ⊆ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴))
10		id 22	. . . . . . 7 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → 𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)))
11	9, 10	sseqtrrid 3965	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (1^st ‘𝐴) ⊆ 𝑥)
12		vex 3433	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	elpw2 5275	. . . . . 6 ⊢ ((1^st ‘𝐴) ∈ 𝒫 𝑥 ↔ (1^st ‘𝐴) ⊆ 𝑥)
14	11, 13	sylibr 234	. . . . 5 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (1^st ‘𝐴) ∈ 𝒫 𝑥)
15		ssun2 4119	. . . . . . 7 ⊢ (2^nd ‘𝐴) ⊆ ((1^st ‘𝐴) ∪ (2^nd ‘𝐴))
16	15, 10	sseqtrrid 3965	. . . . . 6 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (2^nd ‘𝐴) ⊆ 𝑥)
17	12	elpw2 5275	. . . . . 6 ⊢ ((2^nd ‘𝐴) ∈ 𝒫 𝑥 ↔ (2^nd ‘𝐴) ⊆ 𝑥)
18	16, 17	sylibr 234	. . . . 5 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → (2^nd ‘𝐴) ∈ 𝒫 𝑥)
19	14, 18	jca 511	. . . 4 ⊢ (𝑥 = ((1^st ‘𝐴) ∪ (2^nd ‘𝐴)) → ((1^st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2^nd ‘𝐴) ∈ 𝒫 𝑥))
20	8, 19	jctird 526	. . 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 1839	. 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 1950	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 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ∪ cun 3887 ⊆ wss 3889 𝒫 cpw 4541 ‘cfv 6498 1^st c1st 7940 2^nd c2nd 7941 Pgcpg 50184

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-sb 2069 df-clab 2715 df-cleq 2728 df-clel 2811 df-ne 2933 df-rab 3390 df-v 3431 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-pw 4543 df-sn 4568 df-pr 4570 df-uni 4851 df-iota 6454 df-fv 6506

This theorem is referenced by: elpg 50189

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