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Mirrors > Home > ILE Home > Th. List > absneu | 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 3618 | . . . . 5 ⊢ (𝑦 = 𝐴 → {𝑦} = {𝐴}) | |
2 | 1 | eqeq2d 2201 | . . . 4 ⊢ (𝑦 = 𝐴 → ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝐴})) |
3 | 2 | spcegv 2840 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∣ 𝜑} = {𝐴} → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})) |
4 | 3 | imp 124 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
5 | euabsn2 3676 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
6 | 4, 5 | sylibr 134 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∣ 𝜑} = {𝐴}) → ∃!𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∃wex 1503 ∃!weu 2038 ∈ wcel 2160 {cab 2175 {csn 3607 |
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-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-sn 3613 |
This theorem is referenced by: rabsneu 3680 |
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