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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 10977 | . . . 4 ⊢ P ∈ V | |
2 | 1, 1 | xpex 7736 | . . 3 ⊢ (P × P) ∈ V |
3 | 2, 2 | xpex 7736 | . 2 ⊢ ((P × P) × (P × P)) ∈ V |
4 | df-enr 11046 | . . 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 5766 | . . 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 4015 | . 2 ⊢ ~R ⊆ ((P × P) × (P × P)) |
7 | 3, 6 | ssexi 5321 | 1 ⊢ ~R ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3474 ⟨cop 4633 {copab 5209 × cxp 5673 (class class class)co 7405 Pcnp 10850 +P cpp 10852 ~R cer 10855 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-om 7852 df-ni 10863 df-nq 10903 df-np 10972 df-enr 11046 |
This theorem is referenced by: addsrpr 11066 mulsrpr 11067 ltsrpr 11068 0r 11071 1sr 11072 m1r 11073 addclsr 11074 mulclsr 11075 recexsrlem 11094 |
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