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| Mirrors > Home > MPE Home > Th. List > enrex | Structured version Visualization version GIF version | ||
| Description: The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| enrex | ⊢ ~R ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | npex 11026 | . . . 4 ⊢ P ∈ V | |
| 2 | 1, 1 | xpex 7773 | . . 3 ⊢ (P × P) ∈ V |
| 3 | 2, 2 | xpex 7773 | . 2 ⊢ ((P × P) × (P × P)) ∈ V |
| 4 | df-enr 11095 | . . 3 ⊢ ~R = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))} | |
| 5 | opabssxp 5778 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))} ⊆ ((P × P) × (P × P)) | |
| 6 | 4, 5 | eqsstri 4030 | . 2 ⊢ ~R ⊆ ((P × P) × (P × P)) |
| 7 | 3, 6 | ssexi 5322 | 1 ⊢ ~R ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 〈cop 4632 {copab 5205 × cxp 5683 (class class class)co 7431 Pcnp 10899 +P cpp 10901 ~R cer 10904 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-om 7888 df-ni 10912 df-nq 10952 df-np 11021 df-enr 11095 |
| This theorem is referenced by: addsrpr 11115 mulsrpr 11116 ltsrpr 11117 0r 11120 1sr 11121 m1r 11122 addclsr 11123 mulclsr 11124 recexsrlem 11143 |
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