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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 9846 | . . . 4 ⊢ P ∈ V | |
2 | 1, 1 | xpex 7004 | . . 3 ⊢ (P × P) ∈ V |
3 | 2, 2 | xpex 7004 | . 2 ⊢ ((P × P) × (P × P)) ∈ V |
4 | df-enr 9915 | . . 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 5227 | . . 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 3668 | . 2 ⊢ ~R ⊆ ((P × P) × (P × P)) |
7 | 3, 6 | ssexi 4836 | 1 ⊢ ~R ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1523 ∃wex 1744 ∈ wcel 2030 Vcvv 3231 〈cop 4216 {copab 4745 × cxp 5141 (class class class)co 6690 Pcnp 9719 +P cpp 9721 ~R cer 9724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-tr 4786 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-om 7108 df-ni 9732 df-nq 9772 df-np 9841 df-enr 9915 |
This theorem is referenced by: addsrpr 9934 mulsrpr 9935 ltsrpr 9936 0r 9939 1sr 9940 m1r 9941 addclsr 9942 mulclsr 9943 recexsrlem 9962 |
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