Theorem elq 6203

Description: Membership in the set of rationals.

Assertion

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

Distinct variable group:   x,y,A

Proof of Theorem elq

Step	Hyp	Ref	Expression
1		df-q 6202	. . 3 ⊢ ℚ = {z∣∃x ∈ ℤ ∃y ∈ ℕ z = (x / y)}
2	1	eleq2i 1535	. 2 ⊢ (A ∈ ℚ ↔ A ∈ {z∣∃x ∈ ℤ ∃y ∈ ℕ z = (x / y)})
3		oprex 3974	. . . . . . . 8 ⊢ (x / y) ∈ V
4		eleq1 1531	. . . . . . . 8 ⊢ (A = (x / y) → (A ∈ V ↔ (x / y) ∈ V))
5	3, 4	mpbiri 194	. . . . . . 7 ⊢ (A = (x / y) → A ∈ V)
6	5	a1i 8	. . . . . 6 ⊢ (y ∈ ℕ → (A = (x / y) → A ∈ V))
7	6	r19.23aiv 1740	. . . . 5 ⊢ (∃y ∈ ℕ A = (x / y) → A ∈ V)
8	7	a1i 8	. . . 4 ⊢ (x ∈ ℤ → (∃y ∈ ℕ A = (x / y) → A ∈ V))
9	8	r19.23aiv 1740	. . 3 ⊢ (∃x ∈ ℤ ∃y ∈ ℕ A = (x / y) → A ∈ V)
10		eqeq1 1478	. . . 4 ⊢ (z = A → (z = (x / y) ↔ A = (x / y)))
11	10	2rexbidv 1678	. . 3 ⊢ (z = A → (∃x ∈ ℤ ∃y ∈ ℕ z = (x / y) ↔ ∃x ∈ ℤ ∃y ∈ ℕ A = (x / y)))
12	9, 11	elab3 1899	. 2 ⊢ (A ∈ {z∣∃x ∈ ℤ ∃y ∈ ℕ z = (x / y)} ↔ ∃x ∈ ℤ ∃y ∈ ℕ A = (x / y))
13	2, 12	bitr 173	1 ⊢ (A ∈ ℚ ↔ ∃x ∈ ℤ ∃y ∈ ℕ A = (x / y))

Syntax hints:   → wi 3   ↔ wb 146   = wceq 954   ∈ wcel 956  {cab 1461  ∃wrex 1643  Vcvv 1807  (class class class)co 3954   / cdiv 5274  ℕcn 5276  ℤcz 5278  ℚcq 5279

This theorem is referenced by:  znq 6204  qret 6205  zqt 6206  qaddclt 6215  qnegclt 6216  qmulclt 6217  qrecclt 6219  sqr2irr 6667  eirr 7343  qnnen 7454  ipasslem5 8438

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-uni 2499  df-fv 3193  df-opr 3956  df-q 6202

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