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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 1368 | . 2 ⊢ ⊤ | |
2 | reuhyp.1 | . . . 4 ⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶) | |
3 | 2 | adantl 277 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶) |
4 | reuhyp.2 | . . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) | |
5 | 4 | 3adant1 1017 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) |
6 | 3, 5 | reuhypd 4489 | . 2 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
7 | 1, 6 | mpan 424 | 1 ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ⊤wtru 1365 ∈ wcel 2160 ∃!wreu 2470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-reu 2475 df-v 2754 |
This theorem is referenced by: (None) |
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