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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 2046 | . 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | |
2 | df-reu 2423 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | df-rex 2422 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
4 | df-rmo 2424 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | 3, 4 | anbi12i 455 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) |
6 | 1, 2, 5 | 3bitr4i 211 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1468 ∈ wcel 1480 ∃!weu 1999 ∃*wmo 2000 ∃wrex 2417 ∃!wreu 2418 ∃*wrmo 2419 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-rex 2422 df-reu 2423 df-rmo 2424 |
This theorem is referenced by: reurex 2644 reurmo 2645 reu4 2878 reueq 2883 reusv1 4379 fncnv 5189 moriotass 5758 supeuti 6881 infeuti 6916 lteupri 7432 elrealeu 7644 rereceu 7704 exbtwnz 10035 rersqreu 10807 divalglemeunn 11625 divalglemeuneg 11627 bezoutlemeu 11702 pw2dvdseu 11853 dedekindeu 12780 dedekindicclemicc 12789 |
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