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Mirrors > Home > ILE Home > Th. List > reuhyp | GIF version |
Description: A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.) |
Ref | Expression |
---|---|
reuhyp.1 | ⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶) |
reuhyp.2 | ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) |
Ref | Expression |
---|---|
reuhyp | ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1247 | . 2 ⊢ ⊤ | |
2 | reuhyp.1 | . . . 4 ⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶) | |
3 | 2 | adantl 262 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶) |
4 | reuhyp.2 | . . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) | |
5 | 4 | 3adant1 922 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) |
6 | 3, 5 | reuhypd 4203 | . 2 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
7 | 1, 6 | mpan 400 | 1 ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ⊤wtru 1244 ∈ wcel 1393 ∃!wreu 2308 |
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 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-reu 2313 df-v 2559 |
This theorem is referenced by: (None) |
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