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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 7068 | . . . 4 ⊢ N ∈ V | |
2 | 1, 1 | xpex 4614 | . . 3 ⊢ (N × N) ∈ V |
3 | 2, 2 | xpex 4614 | . 2 ⊢ ((N × N) × (N × N)) ∈ V |
4 | df-enq 7103 | . . 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 4573 | . . 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 3095 | . 2 ⊢ ~Q ⊆ ((N × N) × (N × N)) |
7 | 3, 6 | ssexi 4026 | 1 ⊢ ~Q ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1314 ∃wex 1451 ∈ wcel 1463 Vcvv 2657 〈cop 3496 {copab 3948 × cxp 4497 (class class class)co 5728 Ncnpi 7028 ·N cmi 7030 ~Q ceq 7035 |
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 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-iinf 4462 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-opab 3950 df-iom 4465 df-xp 4505 df-ni 7060 df-enq 7103 |
This theorem is referenced by: 1nq 7122 addpipqqs 7126 mulpipqqs 7129 ordpipqqs 7130 addclnq 7131 mulclnq 7132 dmaddpq 7135 dmmulpq 7136 recexnq 7146 ltexnqq 7164 prarloclemarch 7174 prarloclemarch2 7175 nnnq 7178 nqpnq0nq 7209 prarloclemlt 7249 prarloclemlo 7250 prarloclemcalc 7258 nqprm 7298 |
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