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Mirrors > Home > ILE Home > Th. List > enq0er | GIF version |
Description: The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.) |
Ref | Expression |
---|---|
enq0er | ⊢ ~Q0 Er (ω × N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-enq0 7256 | . . . . 5 ⊢ ~Q0 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (ω × N) ∧ 𝑦 ∈ (ω × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 ·o 𝑢) = (𝑤 ·o 𝑣)))} | |
2 | 1 | relopabi 4673 | . . . 4 ⊢ Rel ~Q0 |
3 | 2 | a1i 9 | . . 3 ⊢ (⊤ → Rel ~Q0 ) |
4 | enq0sym 7264 | . . . 4 ⊢ (𝑓 ~Q0 𝑔 → 𝑔 ~Q0 𝑓) | |
5 | 4 | adantl 275 | . . 3 ⊢ ((⊤ ∧ 𝑓 ~Q0 𝑔) → 𝑔 ~Q0 𝑓) |
6 | enq0tr 7266 | . . . 4 ⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → 𝑓 ~Q0 ℎ) | |
7 | 6 | adantl 275 | . . 3 ⊢ ((⊤ ∧ (𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ)) → 𝑓 ~Q0 ℎ) |
8 | enq0ref 7265 | . . . 4 ⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓) | |
9 | 8 | a1i 9 | . . 3 ⊢ (⊤ → (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓)) |
10 | 3, 5, 7, 9 | iserd 6463 | . 2 ⊢ (⊤ → ~Q0 Er (ω × N)) |
11 | 10 | mptru 1341 | 1 ⊢ ~Q0 Er (ω × N) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1332 ⊤wtru 1333 ∃wex 1469 ∈ wcel 1481 〈cop 3535 class class class wbr 3937 ωcom 4512 × cxp 4545 Rel wrel 4552 (class class class)co 5782 ·o comu 6319 Er wer 6434 Ncnpi 7104 ~Q0 ceq0 7118 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-oadd 6325 df-omul 6326 df-er 6437 df-ni 7136 df-enq0 7256 |
This theorem is referenced by: enq0eceq 7269 nqnq0pi 7270 mulcanenq0ec 7277 nnnq0lem1 7278 addnq0mo 7279 mulnq0mo 7280 |
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