Theorem elpglem2 42455

Description: Lemma for elpg 42457. (Contributed by Emmett Weisz, 28-Aug-2021.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem elpglem2

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . 5 ⊢ (1st ‘𝐴) ∈ V
2		fvex 6201	. . . . 5 ⊢ (2nd ‘𝐴) ∈ V
3	1, 2	unex 6956	. . . 4 ⊢ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ∈ V
4	3	isseti 3209	. . 3 ⊢ ∃𝑥 𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴))
5		sseq1 3626	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ⊆ Pg))
6		unss 3787	. . . . . 6 ⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) ↔ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ⊆ Pg)
7	5, 6	syl6bbr 278	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)))
8	7	biimprd 238	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → 𝑥 ⊆ Pg))
9		ssun1 3776	. . . . . . 7 ⊢ (1st ‘𝐴) ⊆ ((1st ‘𝐴) ∪ (2nd ‘𝐴))
10		id 22	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → 𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)))
11	9, 10	syl5sseqr 3654	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (1st ‘𝐴) ⊆ 𝑥)
12		vex 3203	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	elpw2 4828	. . . . . 6 ⊢ ((1st ‘𝐴) ∈ 𝒫 𝑥 ↔ (1st ‘𝐴) ⊆ 𝑥)
14	11, 13	sylibr 224	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (1st ‘𝐴) ∈ 𝒫 𝑥)
15		ssun2 3777	. . . . . . 7 ⊢ (2nd ‘𝐴) ⊆ ((1st ‘𝐴) ∪ (2nd ‘𝐴))
16	15, 10	syl5sseqr 3654	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (2nd ‘𝐴) ⊆ 𝑥)
17	12	elpw2 4828	. . . . . 6 ⊢ ((2nd ‘𝐴) ∈ 𝒫 𝑥 ↔ (2nd ‘𝐴) ⊆ 𝑥)
18	16, 17	sylibr 224	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (2nd ‘𝐴) ∈ 𝒫 𝑥)
19	14, 18	jca 554	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))
20	8, 19	jctird 567	. . 3 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))))
21	4, 20	eximii 1764	. 2 ⊢ ∃𝑥(((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))
22	21	19.37iv 1911	1 ⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ∪ cun 3572   ⊆ wss 3574  𝒫 cpw 4158  ‘cfv 5888  1^st c1st 7166  2^nd c2nd 7167  Pgcpg 42452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851  df-fv 5896

This theorem is referenced by:  elpg  42457