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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 4820. (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 1479 | . 2 ⊢ ⊤ | |
2 | reuhyp.1 | . . . 4 ⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶) | |
3 | 2 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶) |
4 | reuhyp.2 | . . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) | |
5 | 4 | 3adant1 1072 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵)) |
6 | 3, 5 | reuhypd 4821 | . 2 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
7 | 1, 6 | mpan 702 | 1 ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ∃!wreu 2898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-reu 2903 df-v 3175 |
This theorem is referenced by: riotaneg 10879 zriotaneg 11367 zmax 11661 rebtwnz 11663 |
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