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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 7244 | . . . 4 ⊢ N ∈ V | |
2 | 1, 1 | xpex 4713 | . . 3 ⊢ (N × N) ∈ V |
3 | 2, 2 | xpex 4713 | . 2 ⊢ ((N × N) × (N × N)) ∈ V |
4 | df-enq 7279 | . . 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 4672 | . . 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 3169 | . 2 ⊢ ~Q ⊆ ((N × N) × (N × N)) |
7 | 3, 6 | ssexi 4114 | 1 ⊢ ~Q ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ∃wex 1479 ∈ wcel 2135 Vcvv 2721 〈cop 3573 {copab 4036 × cxp 4596 (class class class)co 5836 Ncnpi 7204 ·N cmi 7206 ~Q ceq 7211 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-opab 4038 df-iom 4562 df-xp 4604 df-ni 7236 df-enq 7279 |
This theorem is referenced by: 1nq 7298 addpipqqs 7302 mulpipqqs 7305 ordpipqqs 7306 addclnq 7307 mulclnq 7308 dmaddpq 7311 dmmulpq 7312 recexnq 7322 ltexnqq 7340 prarloclemarch 7350 prarloclemarch2 7351 nnnq 7354 nqpnq0nq 7385 prarloclemlt 7425 prarloclemlo 7426 prarloclemcalc 7434 nqprm 7474 |
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