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Mirrors > Home > MPE Home > Th. List > reuhyp | Structured version Visualization version GIF version |
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4894. (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 1487 | . 2 ⊢ ⊤ | |
2 | reuhyp.1 | . . . 4 ⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶) | |
3 | 2 | adantl 482 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶) |
4 | reuhyp.2 | . . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) | |
5 | 4 | 3adant1 1079 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) |
6 | 3, 5 | reuhypd 4895 | . 2 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
7 | 1, 6 | mpan 706 | 1 ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ∃!wreu 2914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-reu 2919 df-v 3202 |
This theorem is referenced by: riotaneg 11002 zriotaneg 11491 zmax 11785 rebtwnz 11787 |
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