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Mirrors > Home > MPE Home > Th. List > enqex | Structured version Visualization version GIF version |
Description: The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
enqex | ⊢ ~Q ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | niex 10292 | . . . 4 ⊢ N ∈ V | |
2 | 1, 1 | xpex 7456 | . . 3 ⊢ (N × N) ∈ V |
3 | 2, 2 | xpex 7456 | . 2 ⊢ ((N × N) × (N × N)) ∈ V |
4 | df-enq 10322 | . . 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 5607 | . . 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 3949 | . 2 ⊢ ~Q ⊆ ((N × N) × (N × N)) |
7 | 3, 6 | ssexi 5190 | 1 ⊢ ~Q ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 〈cop 4531 {copab 5092 × cxp 5517 (class class class)co 7135 Ncnpi 10255 ·N cmi 10257 ~Q ceq 10262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-om 7561 df-ni 10283 df-enq 10322 |
This theorem is referenced by: (None) |
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