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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 1947 | . 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | |
2 | df-reu 2313 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | df-rex 2312 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
4 | df-rmo 2314 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | 3, 4 | anbi12i 433 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) |
6 | 1, 2, 5 | 3bitr4i 201 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∃wex 1381 ∈ wcel 1393 ∃!weu 1900 ∃*wmo 1901 ∃wrex 2307 ∃!wreu 2308 ∃*wrmo 2309 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-rex 2312 df-reu 2313 df-rmo 2314 |
This theorem is referenced by: reurex 2523 reurmo 2524 reu4 2735 reueq 2738 reusv1 4190 fncnv 4965 moriotass 5496 lteupri 6715 elrealeu 6906 rereceu 6963 qbtwnz 9106 rersqreu 9626 |
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