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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 4589 | . . . 4 ⊢ ω ∈ V | |
2 | niex 7299 | . . . 4 ⊢ N ∈ V | |
3 | 1, 2 | xpex 4738 | . . 3 ⊢ (ω × N) ∈ V |
4 | 3, 3 | xpex 4738 | . 2 ⊢ ((ω × N) × (ω × N)) ∈ V |
5 | df-enq0 7411 | . . 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 4697 | . . 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 3187 | . 2 ⊢ ~Q0 ⊆ ((ω × N) × (ω × N)) |
8 | 4, 7 | ssexi 4138 | 1 ⊢ ~Q0 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 Vcvv 2737 〈cop 3594 {copab 4060 ωcom 4586 × cxp 4621 (class class class)co 5869 ·o comu 6409 Ncnpi 7259 ~Q0 ceq0 7273 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-iinf 4584 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-opab 4062 df-iom 4587 df-xp 4629 df-ni 7291 df-enq0 7411 |
This theorem is referenced by: nqnq0 7428 addnnnq0 7436 mulnnnq0 7437 addclnq0 7438 mulclnq0 7439 prarloclemcalc 7489 |
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