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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 7340 | . . . 4 ⊢ N ∈ V | |
2 | 1, 1 | xpex 4759 | . . 3 ⊢ (N × N) ∈ V |
3 | 2, 2 | xpex 4759 | . 2 ⊢ ((N × N) × (N × N)) ∈ V |
4 | df-enq 7375 | . . 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 4718 | . . 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 3202 | . 2 ⊢ ~Q ⊆ ((N × N) × (N × N)) |
7 | 3, 6 | ssexi 4156 | 1 ⊢ ~Q ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 ∃wex 1503 ∈ wcel 2160 Vcvv 2752 〈cop 3610 {copab 4078 × cxp 4642 (class class class)co 5895 Ncnpi 7300 ·N cmi 7302 ~Q ceq 7307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-opab 4080 df-iom 4608 df-xp 4650 df-ni 7332 df-enq 7375 |
This theorem is referenced by: 1nq 7394 addpipqqs 7398 mulpipqqs 7401 ordpipqqs 7402 addclnq 7403 mulclnq 7404 dmaddpq 7407 dmmulpq 7408 recexnq 7418 ltexnqq 7436 prarloclemarch 7446 prarloclemarch2 7447 nnnq 7450 nqpnq0nq 7481 prarloclemlt 7521 prarloclemlo 7522 prarloclemcalc 7530 nqprm 7570 |
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