Theorem elpglem2 43555

Description: Lemma for elpg 43557. (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 6459	. . . . 5 ⊢ (1st ‘𝐴) ∈ V
2		fvex 6459	. . . . 5 ⊢ (2nd ‘𝐴) ∈ V
3	1, 2	unex 7233	. . . 4 ⊢ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ∈ V
4	3	isseti 3410	. . 3 ⊢ ∃𝑥 𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴))
5		sseq1 3844	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ⊆ Pg))
6		unss 4009	. . . . . 6 ⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) ↔ ((1st ‘𝐴) ∪ (2nd ‘𝐴)) ⊆ Pg)
7	5, 6	syl6bbr 281	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (𝑥 ⊆ Pg ↔ ((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg)))
8	7	biimprd 240	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → 𝑥 ⊆ Pg))
9		ssun1 3998	. . . . . . 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 3872	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (1st ‘𝐴) ⊆ 𝑥)
12		vex 3400	. . . . . . 7 ⊢ 𝑥 ∈ V
13	12	elpw2 5062	. . . . . 6 ⊢ ((1st ‘𝐴) ∈ 𝒫 𝑥 ↔ (1st ‘𝐴) ⊆ 𝑥)
14	11, 13	sylibr 226	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (1st ‘𝐴) ∈ 𝒫 𝑥)
15		ssun2 3999	. . . . . . 7 ⊢ (2nd ‘𝐴) ⊆ ((1st ‘𝐴) ∪ (2nd ‘𝐴))
16	15, 10	syl5sseqr 3872	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (2nd ‘𝐴) ⊆ 𝑥)
17	12	elpw2 5062	. . . . . 6 ⊢ ((2nd ‘𝐴) ∈ 𝒫 𝑥 ↔ (2nd ‘𝐴) ⊆ 𝑥)
18	16, 17	sylibr 226	. . . . 5 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → (2nd ‘𝐴) ∈ 𝒫 𝑥)
19	14, 18	jca 507	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) ∪ (2nd ‘𝐴)) → ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥))
20	8, 19	jctird 522	. . 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 1880	. 2 ⊢ ∃𝑥(((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → (𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))
22	21	19.37iv 1991	1 ⊢ (((1st ‘𝐴) ⊆ Pg ∧ (2nd ‘𝐴) ⊆ Pg) → ∃𝑥(𝑥 ⊆ Pg ∧ ((1st ‘𝐴) ∈ 𝒫 𝑥 ∧ (2nd ‘𝐴) ∈ 𝒫 𝑥)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601  ∃wex 1823   ∈ wcel 2106   ∪ cun 3789   ⊆ wss 3791  𝒫 cpw 4378  ‘cfv 6135  1^st c1st 7443  2^nd c2nd 7444  Pgcpg 43552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-pw 4380  df-sn 4398  df-pr 4400  df-uni 4672  df-iota 6099  df-fv 6143

This theorem is referenced by:  elpg  43557