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Mirrors > Home > ILE Home > Th. List > enq0ex | GIF version |
Description: The equivalence relation for positive fractions exists. (Contributed by Jim Kingdon, 18-Nov-2019.) |
Ref | Expression |
---|---|
enq0ex | ⊢ ~Q0 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 4445 | . . . 4 ⊢ ω ∈ V | |
2 | niex 7021 | . . . 4 ⊢ N ∈ V | |
3 | 1, 2 | xpex 4592 | . . 3 ⊢ (ω × N) ∈ V |
4 | 3, 3 | xpex 4592 | . 2 ⊢ ((ω × N) × (ω × N)) ∈ V |
5 | df-enq0 7133 | . . 3 ⊢ ~Q0 = {〈𝑣, 𝑢〉 ∣ ((𝑣 ∈ (ω × N) ∧ 𝑢 ∈ (ω × N)) ∧ ∃𝑥∃𝑦∃𝑧∃𝑤((𝑣 = 〈𝑥, 𝑦〉 ∧ 𝑢 = 〈𝑧, 𝑤〉) ∧ (𝑥 ·o 𝑤) = (𝑦 ·o 𝑧)))} | |
6 | opabssxp 4551 | . . 3 ⊢ {〈𝑣, 𝑢〉 ∣ ((𝑣 ∈ (ω × N) ∧ 𝑢 ∈ (ω × N)) ∧ ∃𝑥∃𝑦∃𝑧∃𝑤((𝑣 = 〈𝑥, 𝑦〉 ∧ 𝑢 = 〈𝑧, 𝑤〉) ∧ (𝑥 ·o 𝑤) = (𝑦 ·o 𝑧)))} ⊆ ((ω × N) × (ω × N)) | |
7 | 5, 6 | eqsstri 3079 | . 2 ⊢ ~Q0 ⊆ ((ω × N) × (ω × N)) |
8 | 4, 7 | ssexi 4006 | 1 ⊢ ~Q0 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1299 ∃wex 1436 ∈ wcel 1448 Vcvv 2641 〈cop 3477 {copab 3928 ωcom 4442 × cxp 4475 (class class class)co 5706 ·o comu 6241 Ncnpi 6981 ~Q0 ceq0 6995 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-opab 3930 df-iom 4443 df-xp 4483 df-ni 7013 df-enq0 7133 |
This theorem is referenced by: nqnq0 7150 addnnnq0 7158 mulnnnq0 7159 addclnq0 7160 mulclnq0 7161 prarloclemcalc 7211 |
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