Theorem elq 4629

Description: Membership in the set of rationals.

Assertion

Ref	Expression
elq	⊢ (A ∈ ℚ ↔ ∃x ∈ ℤ ∃y ∈ ℕ A = (x / y))

Distinct variable group(s):   x,y,A

Proof of Theorem elq

Step	Hyp	Ref	Expression
1		df-q 4628	. . 3 ⊢ ℚ = {z∣∃x ∈ ℤ ∃y ∈ ℕ z = (x / y)}
2	1	eleq2i 1153	. 2 ⊢ (A ∈ ℚ ↔ A ∈ {z∣∃x ∈ ℤ ∃y ∈ ℕ z = (x / y)})
3		oprex 3018	. . . . . . . 8 ⊢ (x / y) ∈ V
4		eleq1 1149	. . . . . . . 8 ⊢ (A = (x / y) → (A ∈ V ↔ (x / y) ∈ V))
5	3, 4	mpbiri 169	. . . . . . 7 ⊢ (A = (x / y) → A ∈ V)
6	5	a1i 7	. . . . . 6 ⊢ (y ∈ ℕ → (A = (x / y) → A ∈ V))
7	6	r19.23aiv 1284	. . . . 5 ⊢ (∃y ∈ ℕ A = (x / y) → A ∈ V)
8	7	a1i 7	. . . 4 ⊢ (x ∈ ℤ → (∃y ∈ ℕ A = (x / y) → A ∈ V))
9	8	r19.23aiv 1284	. . 3 ⊢ (∃x ∈ ℤ ∃y ∈ ℕ A = (x / y) → A ∈ V)
10		cleq1 1107	. . . . 5 ⊢ (z = A → (z = (x / y) ↔ A = (x / y)))
11	10	birexdv 1220	. . . 4 ⊢ (z = A → (∃y ∈ ℕ z = (x / y) ↔ ∃y ∈ ℕ A = (x / y)))
12	11	birexdv 1220	. . 3 ⊢ (z = A → (∃x ∈ ℤ ∃y ∈ ℕ z = (x / y) ↔ ∃x ∈ ℤ ∃y ∈ ℕ A = (x / y)))
13	9, 12	elab3g 1420	. 2 ⊢ (A ∈ {z∣∃x ∈ ℤ ∃y ∈ ℕ z = (x / y)} ↔ ∃x ∈ ℤ ∃y ∈ ℕ A = (x / y))
14	2, 13	bitr 151	1 ⊢ (A ∈ ℚ ↔ ∃x ∈ ℤ ∃y ∈ ℕ A = (x / y))

Syntax hints:   → wi 2   ↔ wb 127  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  (class class class)co 3001   / cdiv 4091  ℕcn 4093  ℤcz 4095  ℚcq 4096

This theorem is referenced by:  znq 4630  qret 4631  zqt 4632  qaddclt 4642  qnegclt 4643  qmulclt 4644  qrecclt 4646  sqr2irr 4782  qnnen 4931

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-uni 1920  df-fv 2438  df-opr 3003  df-q 4628

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