ssetkex - New Foundations Explorer

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Theorem ssetkex 4295

Description: The Kuratowski subset relationship is a set. (Contributed by SF, 13-Jan-2015.)

**Assertion**
Ref	Expression
ssetkex	⊢ S_k ∈ V

Proof of Theorem ssetkex Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-sset 4083	. 2 ⊢ ∃x∀y∀z(⟪y, z⟫ ∈ x ↔ ∀w(w ∈ y → w ∈ z))
2		inss1 3476	. . . . . . 7 ⊢ ((V ×_k V) ∩ x) ⊆ (V ×_k V)
3		ssetkssvvk 4279	. . . . . . 7 ⊢ S_k ⊆ (V ×_k V)
4		eqrelk 4213	. . . . . . 7 ⊢ ((((V ×_k V) ∩ x) ⊆ (V ×_k V) ∧ S_k ⊆ (V ×_k V)) → (((V ×_k V) ∩ x) = S_k ↔ ∀y∀z(⟪y, z⟫ ∈ ((V ×_k V) ∩ x) ↔ ⟪y, z⟫ ∈ S_k )))
5	2, 3, 4	mp2an 653	. . . . . 6 ⊢ (((V ×_k V) ∩ x) = S_k ↔ ∀y∀z(⟪y, z⟫ ∈ ((V ×_k V) ∩ x) ↔ ⟪y, z⟫ ∈ S_k ))
6		vex 2863	. . . . . . . . . 10 ⊢ y ∈ V
7		vex 2863	. . . . . . . . . 10 ⊢ z ∈ V
8	6, 7	opkelxpk 4249	. . . . . . . . . 10 ⊢ (⟪y, z⟫ ∈ (V ×_k V) ↔ (y ∈ V ∧ z ∈ V))
9	6, 7, 8	mpbir2an 886	. . . . . . . . 9 ⊢ ⟪y, z⟫ ∈ (V ×_k V)
10		elin 3220	. . . . . . . . 9 ⊢ (⟪y, z⟫ ∈ ((V ×_k V) ∩ x) ↔ (⟪y, z⟫ ∈ (V ×_k V) ∧ ⟪y, z⟫ ∈ x))
11	9, 10	mpbiran 884	. . . . . . . 8 ⊢ (⟪y, z⟫ ∈ ((V ×_k V) ∩ x) ↔ ⟪y, z⟫ ∈ x)
12		opkelssetkg 4269	. . . . . . . . . 10 ⊢ ((y ∈ V ∧ z ∈ V) → (⟪y, z⟫ ∈ S_k ↔ y ⊆ z))
13	6, 7, 12	mp2an 653	. . . . . . . . 9 ⊢ (⟪y, z⟫ ∈ S_k ↔ y ⊆ z)
14		dfss2 3263	. . . . . . . . 9 ⊢ (y ⊆ z ↔ ∀w(w ∈ y → w ∈ z))
15	13, 14	bitri 240	. . . . . . . 8 ⊢ (⟪y, z⟫ ∈ S_k ↔ ∀w(w ∈ y → w ∈ z))
16	11, 15	bibi12i 306	. . . . . . 7 ⊢ ((⟪y, z⟫ ∈ ((V ×_k V) ∩ x) ↔ ⟪y, z⟫ ∈ S_k ) ↔ (⟪y, z⟫ ∈ x ↔ ∀w(w ∈ y → w ∈ z)))
17	16	2albii 1567	. . . . . 6 ⊢ (∀y∀z(⟪y, z⟫ ∈ ((V ×_k V) ∩ x) ↔ ⟪y, z⟫ ∈ S_k ) ↔ ∀y∀z(⟪y, z⟫ ∈ x ↔ ∀w(w ∈ y → w ∈ z)))
18	5, 17	bitri 240	. . . . 5 ⊢ (((V ×_k V) ∩ x) = S_k ↔ ∀y∀z(⟪y, z⟫ ∈ x ↔ ∀w(w ∈ y → w ∈ z)))
19	18	biimpri 197	. . . 4 ⊢ (∀y∀z(⟪y, z⟫ ∈ x ↔ ∀w(w ∈ y → w ∈ z)) → ((V ×_k V) ∩ x) = S_k )
20		vvex 4110	. . . . . 6 ⊢ V ∈ V
21		xpkvexg 4286	. . . . . 6 ⊢ (V ∈ V → (V ×_k V) ∈ V)
22	20, 21	ax-mp 5	. . . . 5 ⊢ (V ×_k V) ∈ V
23		vex 2863	. . . . 5 ⊢ x ∈ V
24	22, 23	inex 4106	. . . 4 ⊢ ((V ×_k V) ∩ x) ∈ V
25	19, 24	syl6eqelr 2442	. . 3 ⊢ (∀y∀z(⟪y, z⟫ ∈ x ↔ ∀w(w ∈ y → w ∈ z)) → S_k ∈ V)
26	25	exlimiv 1634	. 2 ⊢ (∃x∀y∀z(⟪y, z⟫ ∈ x ↔ ∀w(w ∈ y → w ∈ z)) → S_k ∈ V)
27	1, 26	ax-mp 5	1 ⊢ S_k ∈ V

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 176 ∀wal 1540 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2860 ∩ cin 3209 ⊆ wss 3258 ⟪copk 4058 ×_k cxpk 4175 S_k cssetk 4184

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-xp 4080 ax-sset 4083 ax-sn 4088

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-v 2862 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-ss 3260 df-nul 3552 df-sn 3742 df-pr 3743 df-opk 4059 df-xpk 4186 df-ssetk 4194

This theorem is referenced by: imagekexg 4312 idkex 4315 uniexg 4317 intexg 4320 setswithex 4323 pwexg 4329 addcexlem 4383 nncex 4397 nnsucelrlem1 4425 nndisjeq 4430 ltfinex 4465 ssfin 4471 ncfinraiselem2 4481 ncfinlowerlem1 4483 tfinrelkex 4488 evenfinex 4504 oddfinex 4505 evenodddisjlem1 4516 nnadjoinlem1 4520 nnpweqlem1 4523 srelkex 4526 tfinnnlem1 4534 spfinex 4538 opexg 4588 proj2exg 4593 setconslem5 4736 1stex 4740 swapex 4743 ssetex 4745

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