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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 4610 | . . . 4 ⊢ ω ∈ V | |
2 | niex 7342 | . . . 4 ⊢ N ∈ V | |
3 | 1, 2 | xpex 4759 | . . 3 ⊢ (ω × N) ∈ V |
4 | 3, 3 | xpex 4759 | . 2 ⊢ ((ω × N) × (ω × N)) ∈ V |
5 | df-enq0 7454 | . . 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 4718 | . . 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 3202 | . 2 ⊢ ~Q0 ⊆ ((ω × N) × (ω × N)) |
8 | 4, 7 | ssexi 4156 | 1 ⊢ ~Q0 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 ∃wex 1503 ∈ wcel 2160 Vcvv 2752 〈cop 3610 {copab 4078 ωcom 4607 × cxp 4642 (class class class)co 5897 ·o comu 6440 Ncnpi 7302 ~Q0 ceq0 7316 |
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 7334 df-enq0 7454 |
This theorem is referenced by: nqnq0 7471 addnnnq0 7479 mulnnnq0 7480 addclnq0 7481 mulclnq0 7482 prarloclemcalc 7532 |
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