Theorem 0nnq 9947

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

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

Step	Hyp	Ref	Expression
1		0nelxp 5283	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 9935	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3		ssrab2 3834	. . . 4 ⊢ {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))} ⊆ (N × N)
4	2, 3	eqsstri 3782	. . 3 ⊢ Q ⊆ (N × N)
5	4	sseli 3746	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
6	1, 5	mto 188	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2144  ∀wral 3060  {crab 3064  ∅c0 4061   class class class wbr 4784   × cxp 5247  ‘cfv 6031  2^nd c2nd 7313  Ncnpi 9867   <_N clti 9870   ~_Q ceq 9874  Qcnq 9875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-opab 4845  df-xp 5255  df-nq 9935

This theorem is referenced by:  adderpq  9979  mulerpq  9980  addassnq  9981  mulassnq  9982  distrnq  9984  recmulnq  9987  recclnq  9989  ltanq  9994  ltmnq  9995  ltexnq  9998  nsmallnq  10000  ltbtwnnq  10001  ltrnq  10002  prlem934  10056  ltaddpr  10057  ltexprlem2  10060  ltexprlem3  10061  ltexprlem4  10062  ltexprlem6  10064  ltexprlem7  10065  prlem936  10070  reclem2pr  10071