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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 7393 | . . . 4 ⊢ P ∈ V | |
2 | 1, 1 | xpex 4701 | . . 3 ⊢ (P × P) ∈ V |
3 | 2, 2 | xpex 4701 | . 2 ⊢ ((P × P) × (P × P)) ∈ V |
4 | df-enr 7646 | . . 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 4660 | . . 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 3160 | . 2 ⊢ ~R ⊆ ((P × P) × (P × P)) |
7 | 3, 6 | ssexi 4102 | 1 ⊢ ~R ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1335 ∃wex 1472 ∈ wcel 2128 Vcvv 2712 〈cop 3563 {copab 4024 × cxp 4584 (class class class)co 5824 Pcnp 7211 +P cpp 7213 ~R cer 7216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-qs 6486 df-ni 7224 df-nqqs 7268 df-inp 7386 df-enr 7646 |
This theorem is referenced by: addsrpr 7665 mulsrpr 7666 ltsrprg 7667 0r 7670 1sr 7671 m1r 7672 addclsr 7673 mulclsr 7674 recexgt0sr 7693 prsrcl 7704 ltpsrprg 7723 mappsrprg 7724 suplocsrlemb 7726 pitonnlem2 7767 pitonn 7768 pitore 7770 recnnre 7771 |
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