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Mirrors > Home > ILE Home > Th. List > enq0er | GIF version |
Description: The equivalence relation for non-negative 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 6522 | . . . . 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 4463 | . . . 4 ⊢ Rel ~Q0 |
3 | 2 | a1i 9 | . . 3 ⊢ (⊤ → Rel ~Q0 ) |
4 | enq0sym 6530 | . . . 4 ⊢ (𝑓 ~Q0 𝑔 → 𝑔 ~Q0 𝑓) | |
5 | 4 | adantl 262 | . . 3 ⊢ ((⊤ ∧ 𝑓 ~Q0 𝑔) → 𝑔 ~Q0 𝑓) |
6 | enq0tr 6532 | . . . 4 ⊢ ((𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ) → 𝑓 ~Q0 ℎ) | |
7 | 6 | adantl 262 | . . 3 ⊢ ((⊤ ∧ (𝑓 ~Q0 𝑔 ∧ 𝑔 ~Q0 ℎ)) → 𝑓 ~Q0 ℎ) |
8 | enq0ref 6531 | . . . 4 ⊢ (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓) | |
9 | 8 | a1i 9 | . . 3 ⊢ (⊤ → (𝑓 ∈ (ω × N) ↔ 𝑓 ~Q0 𝑓)) |
10 | 3, 5, 7, 9 | iserd 6132 | . 2 ⊢ (⊤ → ~Q0 Er (ω × N)) |
11 | 10 | trud 1252 | 1 ⊢ ~Q0 Er (ω × N) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ⊤wtru 1244 ∃wex 1381 ∈ wcel 1393 〈cop 3378 class class class wbr 3764 ωcom 4313 × cxp 4343 Rel wrel 4350 (class class class)co 5512 ·𝑜 comu 5999 Er wer 6103 Ncnpi 6370 ~Q0 ceq0 6384 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 df-er 6106 df-ni 6402 df-enq0 6522 |
This theorem is referenced by: enq0eceq 6535 nqnq0pi 6536 mulcanenq0ec 6543 nnnq0lem1 6544 addnq0mo 6545 mulnq0mo 6546 |
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