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Mirrors > Home > ILE Home > Th. List > reu5 | GIF version |
Description: Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
reu5 | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eu5 1989 | . 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | |
2 | df-reu 2356 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | df-rex 2355 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
4 | df-rmo 2357 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | 3, 4 | anbi12i 448 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) |
6 | 1, 2, 5 | 3bitr4i 210 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∃wex 1422 ∈ wcel 1434 ∃!weu 1942 ∃*wmo 1943 ∃wrex 2350 ∃!wreu 2351 ∃*wrmo 2352 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 |
This theorem depends on definitions: df-bi 115 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-rex 2355 df-reu 2356 df-rmo 2357 |
This theorem is referenced by: reurex 2568 reurmo 2569 reu4 2787 reueq 2790 reusv1 4210 fncnv 4990 moriotass 5521 supeuti 6456 infeuti 6491 lteupri 6858 elrealeu 7049 rereceu 7106 exbtwnz 9326 rersqreu 10041 divalglemeunn 10454 divalglemeuneg 10456 bezoutlemeu 10529 pw2dvdseu 10679 |
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