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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 10407 | . . . 4 ⊢ P ∈ V | |
2 | 1, 1 | xpex 7475 | . . 3 ⊢ (P × P) ∈ V |
3 | 2, 2 | xpex 7475 | . 2 ⊢ ((P × P) × (P × P)) ∈ V |
4 | df-enr 10476 | . . 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 5642 | . . 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 4000 | . 2 ⊢ ~R ⊆ ((P × P) × (P × P)) |
7 | 3, 6 | ssexi 5225 | 1 ⊢ ~R ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 〈cop 4572 {copab 5127 × cxp 5552 (class class class)co 7155 Pcnp 10280 +P cpp 10282 ~R cer 10285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-tr 5172 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-om 7580 df-ni 10293 df-nq 10333 df-np 10402 df-enr 10476 |
This theorem is referenced by: addsrpr 10496 mulsrpr 10497 ltsrpr 10498 0r 10501 1sr 10502 m1r 10503 addclsr 10504 mulclsr 10505 recexsrlem 10524 |
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