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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 4259 | . . . 4 ⊢ 𝜔 ∈ V | |
2 | niex 6296 | . . . 4 ⊢ N ∈ V | |
3 | 1, 2 | xpex 4396 | . . 3 ⊢ (𝜔 × N) ∈ V |
4 | 3, 3 | xpex 4396 | . 2 ⊢ ((𝜔 × N) × (𝜔 × N)) ∈ V |
5 | df-enq0 6407 | . . 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 4357 | . . 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 2969 | . 2 ⊢ ~Q0 ⊆ ((𝜔 × N) × (𝜔 × N)) |
8 | 4, 7 | ssexi 3886 | 1 ⊢ ~Q0 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1242 ∃wex 1378 ∈ wcel 1390 Vcvv 2551 〈cop 3370 {copab 3808 𝜔com 4256 × cxp 4286 (class class class)co 5455 ·𝑜 comu 5938 Ncnpi 6256 ~Q0 ceq0 6270 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-opab 3810 df-iom 4257 df-xp 4294 df-ni 6288 df-enq0 6407 |
This theorem is referenced by: nqnq0 6424 addnnnq0 6432 mulnnnq0 6433 addclnq0 6434 mulclnq0 6435 prarloclemcalc 6485 |
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