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Mirrors > Home > ILE Home > Th. List > enrex | GIF version |
Description: The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) |
Ref | Expression |
---|---|
enrex | ⊢ ~R ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | npex 7471 | . . . 4 ⊢ P ∈ V | |
2 | 1, 1 | xpex 4741 | . . 3 ⊢ (P × P) ∈ V |
3 | 2, 2 | xpex 4741 | . 2 ⊢ ((P × P) × (P × P)) ∈ V |
4 | df-enr 7724 | . . 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 4700 | . . 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 3187 | . 2 ⊢ ~R ⊆ ((P × P) × (P × P)) |
7 | 3, 6 | ssexi 4141 | 1 ⊢ ~R ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 Vcvv 2737 ⟨cop 3595 {copab 4063 × cxp 4624 (class class class)co 5874 Pcnp 7289 +P cpp 7291 ~R cer 7294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-qs 6540 df-ni 7302 df-nqqs 7346 df-inp 7464 df-enr 7724 |
This theorem is referenced by: addsrpr 7743 mulsrpr 7744 ltsrprg 7745 0r 7748 1sr 7749 m1r 7750 addclsr 7751 mulclsr 7752 recexgt0sr 7771 prsrcl 7782 ltpsrprg 7801 mappsrprg 7802 suplocsrlemb 7804 pitonnlem2 7845 pitonn 7846 pitore 7848 recnnre 7849 |
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