|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > enqex | Structured version Visualization version GIF version | ||
| Description: The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| enqex | ⊢ ~Q ∈ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | niex 10922 | . . . 4 ⊢ N ∈ V | |
| 2 | 1, 1 | xpex 7774 | . . 3 ⊢ (N × N) ∈ V | 
| 3 | 2, 2 | xpex 7774 | . 2 ⊢ ((N × N) × (N × N)) ∈ V | 
| 4 | df-enq 10952 | . . 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 5777 | . . 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 4029 | . 2 ⊢ ~Q ⊆ ((N × N) × (N × N)) | 
| 7 | 3, 6 | ssexi 5321 | 1 ⊢ ~Q ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 Vcvv 3479 〈cop 4631 {copab 5204 × cxp 5682 (class class class)co 7432 Ncnpi 10885 ·N cmi 10887 ~Q ceq 10892 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-om 7889 df-ni 10913 df-enq 10952 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |