Theorem nqnq0 6423

Description: A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.)

Assertion

Ref	Expression
nqnq0	⊢ Q ⊆ Q0

Proof of Theorem nqnq0

Dummy variables v u x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nqqs 6332	. . . . 5 ⊢ Q = ((N × N) / ~Q )
2	1	eleq2i 2101	. . . 4 ⊢ (y ∈ Q ↔ y ∈ ((N × N) / ~Q ))
3		vex 2554	. . . . 5 ⊢ y ∈ V
4	3	elqs 6093	. . . 4 ⊢ (y ∈ ((N × N) / ~Q ) ↔ ∃x ∈ (N × N)y = [x] ~Q )
5		df-rex 2306	. . . 4 ⊢ (∃x ∈ (N × N)y = [x] ~Q ↔ ∃x(x ∈ (N × N) ∧ y = [x] ~Q ))
6	2, 4, 5	3bitri 195	. . 3 ⊢ (y ∈ Q ↔ ∃x(x ∈ (N × N) ∧ y = [x] ~Q ))
7		elxpi 4304	. . . . . . 7 ⊢ (x ∈ (N × N) → ∃u∃v(x = ⟨u, v⟩ ∧ (u ∈ N ∧ v ∈ N)))
8		nqnq0pi 6420	. . . . . . . . . . 11 ⊢ ((u ∈ N ∧ v ∈ N) → [⟨u, v⟩] ~Q0 = [⟨u, v⟩] ~Q )
9	8	adantl 262	. . . . . . . . . 10 ⊢ ((x = ⟨u, v⟩ ∧ (u ∈ N ∧ v ∈ N)) → [⟨u, v⟩] ~Q0 = [⟨u, v⟩] ~Q )
10		eceq1 6077	. . . . . . . . . . . 12 ⊢ (x = ⟨u, v⟩ → [x] ~Q0 = [⟨u, v⟩] ~Q0 )
11		eceq1 6077	. . . . . . . . . . . 12 ⊢ (x = ⟨u, v⟩ → [x] ~Q = [⟨u, v⟩] ~Q )
12	10, 11	eqeq12d 2051	. . . . . . . . . . 11 ⊢ (x = ⟨u, v⟩ → ([x] ~Q0 = [x] ~Q ↔ [⟨u, v⟩] ~Q0 = [⟨u, v⟩] ~Q ))
13	12	adantr 261	. . . . . . . . . 10 ⊢ ((x = ⟨u, v⟩ ∧ (u ∈ N ∧ v ∈ N)) → ([x] ~Q0 = [x] ~Q ↔ [⟨u, v⟩] ~Q0 = [⟨u, v⟩] ~Q ))
14	9, 13	mpbird 156	. . . . . . . . 9 ⊢ ((x = ⟨u, v⟩ ∧ (u ∈ N ∧ v ∈ N)) → [x] ~Q0 = [x] ~Q )
15		pinn 6293	. . . . . . . . . . . . 13 ⊢ (u ∈ N → u ∈ 𝜔)
16		opelxpi 4319	. . . . . . . . . . . . 13 ⊢ ((u ∈ 𝜔 ∧ v ∈ N) → ⟨u, v⟩ ∈ (𝜔 × N))
17	15, 16	sylan 267	. . . . . . . . . . . 12 ⊢ ((u ∈ N ∧ v ∈ N) → ⟨u, v⟩ ∈ (𝜔 × N))
18	17	adantl 262	. . . . . . . . . . 11 ⊢ ((x = ⟨u, v⟩ ∧ (u ∈ N ∧ v ∈ N)) → ⟨u, v⟩ ∈ (𝜔 × N))
19		eleq1 2097	. . . . . . . . . . . 12 ⊢ (x = ⟨u, v⟩ → (x ∈ (𝜔 × N) ↔ ⟨u, v⟩ ∈ (𝜔 × N)))
20	19	adantr 261	. . . . . . . . . . 11 ⊢ ((x = ⟨u, v⟩ ∧ (u ∈ N ∧ v ∈ N)) → (x ∈ (𝜔 × N) ↔ ⟨u, v⟩ ∈ (𝜔 × N)))
21	18, 20	mpbird 156	. . . . . . . . . 10 ⊢ ((x = ⟨u, v⟩ ∧ (u ∈ N ∧ v ∈ N)) → x ∈ (𝜔 × N))
22		enq0ex 6421	. . . . . . . . . . . 12 ⊢ ~Q0 ∈ V
23	22	ecelqsi 6096	. . . . . . . . . . 11 ⊢ (x ∈ (𝜔 × N) → [x] ~Q0 ∈ ((𝜔 × N) / ~Q0 ))
24		df-nq0 6407	. . . . . . . . . . 11 ⊢ Q0 = ((𝜔 × N) / ~Q0 )
25	23, 24	syl6eleqr 2128	. . . . . . . . . 10 ⊢ (x ∈ (𝜔 × N) → [x] ~Q0 ∈ Q0)
26	21, 25	syl 14	. . . . . . . . 9 ⊢ ((x = ⟨u, v⟩ ∧ (u ∈ N ∧ v ∈ N)) → [x] ~Q0 ∈ Q0)
27	14, 26	eqeltrrd 2112	. . . . . . . 8 ⊢ ((x = ⟨u, v⟩ ∧ (u ∈ N ∧ v ∈ N)) → [x] ~Q ∈ Q0)
28	27	exlimivv 1773	. . . . . . 7 ⊢ (∃u∃v(x = ⟨u, v⟩ ∧ (u ∈ N ∧ v ∈ N)) → [x] ~Q ∈ Q0)
29	7, 28	syl 14	. . . . . 6 ⊢ (x ∈ (N × N) → [x] ~Q ∈ Q0)
30	29	adantr 261	. . . . 5 ⊢ ((x ∈ (N × N) ∧ y = [x] ~Q ) → [x] ~Q ∈ Q0)
31		eleq1 2097	. . . . . 6 ⊢ (y = [x] ~Q → (y ∈ Q0 ↔ [x] ~Q ∈ Q0))
32	31	adantl 262	. . . . 5 ⊢ ((x ∈ (N × N) ∧ y = [x] ~Q ) → (y ∈ Q0 ↔ [x] ~Q ∈ Q0))
33	30, 32	mpbird 156	. . . 4 ⊢ ((x ∈ (N × N) ∧ y = [x] ~Q ) → y ∈ Q0)
34	33	exlimiv 1486	. . 3 ⊢ (∃x(x ∈ (N × N) ∧ y = [x] ~Q ) → y ∈ Q0)
35	6, 34	sylbi 114	. 2 ⊢ (y ∈ Q → y ∈ Q0)
36	35	ssriv 2943	1 ⊢ Q ⊆ Q0

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301   ⊆ wss 2911  ⟨cop 3370  𝜔com 4256   × cxp 4286  [cec 6040   / cqs 6041  Ncnpi 6256   ~_Q ceq 6263  Qcnq 6264   ~_Q0 ceq0 6270  Q₀cnq0 6271

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-mi 6290  df-enq 6331  df-nqqs 6332  df-enq0 6406  df-nq0 6407

This theorem is referenced by:  prarloclem5  6482  prarloclemcalc  6484