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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 11029 | . . . 4 ⊢ P ∈ V | |
2 | 1, 1 | xpex 7761 | . . 3 ⊢ (P × P) ∈ V |
3 | 2, 2 | xpex 7761 | . 2 ⊢ ((P × P) × (P × P)) ∈ V |
4 | df-enr 11098 | . . 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 5774 | . . 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 4014 | . 2 ⊢ ~R ⊆ ((P × P) × (P × P)) |
7 | 3, 6 | ssexi 5327 | 1 ⊢ ~R ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3462 〈cop 4639 {copab 5215 × cxp 5680 (class class class)co 7424 Pcnp 10902 +P cpp 10904 ~R cer 10907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-tr 5271 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-om 7877 df-ni 10915 df-nq 10955 df-np 11024 df-enr 11098 |
This theorem is referenced by: addsrpr 11118 mulsrpr 11119 ltsrpr 11120 0r 11123 1sr 11124 m1r 11125 addclsr 11126 mulclsr 11127 recexsrlem 11146 |
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