Theorem enqex 7493

Description: The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.)

Assertion

Ref	Expression
enqex	⊢ ~Q ∈ V

Proof of Theorem enqex

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

Step	Hyp	Ref	Expression
1		niex 7445	. . . 4 ⊢ N ∈ V
2	1, 1	xpex 4798	. . 3 ⊢ (N × N) ∈ V
3	2, 2	xpex 4798	. 2 ⊢ ((N × N) × (N × N)) ∈ V
4		df-enq 7480	. . 3 ⊢ ~Q = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))}
5		opabssxp 4757	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))} ⊆ ((N × N) × (N × N))
6	4, 5	eqsstri 3229	. 2 ⊢ ~Q ⊆ ((N × N) × (N × N))
7	3, 6	ssexi 4190	1 ⊢ ~Q ∈ V

Syntax hints:   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773  ⟨cop 3641  {copab 4112   × cxp 4681  (class class class)co 5957  Ncnpi 7405   ·_N cmi 7407   ~_Q ceq 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-opab 4114  df-iom 4647  df-xp 4689  df-ni 7437  df-enq 7480

This theorem is referenced by:  1nq  7499  addpipqqs  7503  mulpipqqs  7506  ordpipqqs  7507  addclnq  7508  mulclnq  7509  dmaddpq  7512  dmmulpq  7513  recexnq  7523  ltexnqq  7541  prarloclemarch  7551  prarloclemarch2  7552  nnnq  7555  nqpnq0nq  7586  prarloclemlt  7626  prarloclemlo  7627  prarloclemcalc  7635  nqprm  7675