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Mirrors > Home > MPE Home > Th. List > rabsneu | Structured version Visualization version GIF version |
Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
rabsneu | ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴}) → ∃!𝑥 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3429 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
2 | 1 | eqeq1i 2732 | . . 3 ⊢ ({𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴} ↔ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} = {𝐴}) |
3 | absneu 4735 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} = {𝐴}) → ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
4 | 2, 3 | sylan2b 592 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴}) → ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) |
5 | df-reu 3373 | . 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
6 | 4, 5 | sylibr 233 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝐴}) → ∃!𝑥 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃!weu 2557 {cab 2704 ∃!wreu 3370 {crab 3428 {csn 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-reu 3373 df-rab 3429 df-sn 4631 |
This theorem is referenced by: (None) |
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