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Mirrors > Home > ILE Home > Th. List > enqex | GIF version |
Description: The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) |
Ref | Expression |
---|---|
enqex | ⊢ ~Q ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | niex 7088 | . . . 4 ⊢ N ∈ V | |
2 | 1, 1 | xpex 4624 | . . 3 ⊢ (N × N) ∈ V |
3 | 2, 2 | xpex 4624 | . 2 ⊢ ((N × N) × (N × N)) ∈ V |
4 | df-enq 7123 | . . 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 4583 | . . 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 3099 | . 2 ⊢ ~Q ⊆ ((N × N) × (N × N)) |
7 | 3, 6 | ssexi 4036 | 1 ⊢ ~Q ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1316 ∃wex 1453 ∈ wcel 1465 Vcvv 2660 〈cop 3500 {copab 3958 × cxp 4507 (class class class)co 5742 Ncnpi 7048 ·N cmi 7050 ~Q ceq 7055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-opab 3960 df-iom 4475 df-xp 4515 df-ni 7080 df-enq 7123 |
This theorem is referenced by: 1nq 7142 addpipqqs 7146 mulpipqqs 7149 ordpipqqs 7150 addclnq 7151 mulclnq 7152 dmaddpq 7155 dmmulpq 7156 recexnq 7166 ltexnqq 7184 prarloclemarch 7194 prarloclemarch2 7195 nnnq 7198 nqpnq0nq 7229 prarloclemlt 7269 prarloclemlo 7270 prarloclemcalc 7278 nqprm 7318 |
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