Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > enqex | GIF version |
Description: The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) |
Ref | Expression |
---|---|
enqex | ⊢ ~Q ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | niex 7261 | . . . 4 ⊢ N ∈ V | |
2 | 1, 1 | xpex 4724 | . . 3 ⊢ (N × N) ∈ V |
3 | 2, 2 | xpex 4724 | . 2 ⊢ ((N × N) × (N × N)) ∈ V |
4 | df-enq 7296 | . . 3 ⊢ ~Q = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))} | |
5 | opabssxp 4683 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))} ⊆ ((N × N) × (N × N)) | |
6 | 4, 5 | eqsstri 3179 | . 2 ⊢ ~Q ⊆ ((N × N) × (N × N)) |
7 | 3, 6 | ssexi 4125 | 1 ⊢ ~Q ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 ∃wex 1485 ∈ wcel 2141 Vcvv 2730 〈cop 3584 {copab 4047 × cxp 4607 (class class class)co 5850 Ncnpi 7221 ·N cmi 7223 ~Q ceq 7228 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-opab 4049 df-iom 4573 df-xp 4615 df-ni 7253 df-enq 7296 |
This theorem is referenced by: 1nq 7315 addpipqqs 7319 mulpipqqs 7322 ordpipqqs 7323 addclnq 7324 mulclnq 7325 dmaddpq 7328 dmmulpq 7329 recexnq 7339 ltexnqq 7357 prarloclemarch 7367 prarloclemarch2 7368 nnnq 7371 nqpnq0nq 7402 prarloclemlt 7442 prarloclemlo 7443 prarloclemcalc 7451 nqprm 7491 |
Copyright terms: Public domain | W3C validator |