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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 7305 | . . . 4 ⊢ P ∈ V | |
2 | 1, 1 | xpex 4662 | . . 3 ⊢ (P × P) ∈ V |
3 | 2, 2 | xpex 4662 | . 2 ⊢ ((P × P) × (P × P)) ∈ V |
4 | df-enr 7558 | . . 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 4621 | . . 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 3134 | . 2 ⊢ ~R ⊆ ((P × P) × (P × P)) |
7 | 3, 6 | ssexi 4074 | 1 ⊢ ~R ∈ V |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 ∃wex 1469 ∈ wcel 1481 Vcvv 2689 〈cop 3535 {copab 3996 × cxp 4545 (class class class)co 5782 Pcnp 7123 +P cpp 7125 ~R cer 7128 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-qs 6443 df-ni 7136 df-nqqs 7180 df-inp 7298 df-enr 7558 |
This theorem is referenced by: addsrpr 7577 mulsrpr 7578 ltsrprg 7579 0r 7582 1sr 7583 m1r 7584 addclsr 7585 mulclsr 7586 recexgt0sr 7605 prsrcl 7616 ltpsrprg 7635 mappsrprg 7636 suplocsrlemb 7638 pitonnlem2 7679 pitonn 7680 pitore 7682 recnnre 7683 |
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