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Mirrors > Home > MPE Home > Th. List > absneu | Structured version Visualization version GIF version |
Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) |
Ref | Expression |
---|---|
absneu | ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4631 | . . . . 5 ⊢ (𝑦 = 𝐴 → {𝑦} = {𝐴}) | |
2 | 1 | eqeq2d 2735 | . . . 4 ⊢ (𝑦 = 𝐴 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝐴})) |
3 | 2 | spcegv 3579 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∣ 𝜑} = {𝐴} → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})) |
4 | 3 | imp 406 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
5 | euabsn2 4722 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
6 | 4, 5 | sylibr 233 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∃!weu 2554 {cab 2701 {csn 4621 |
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 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-sn 4622 |
This theorem is referenced by: rabsneu 4726 |
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