![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > snriota | GIF version |
Description: A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.) |
Ref | Expression |
---|---|
snriota | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {(℩𝑥 ∈ 𝐴 𝜑)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 2475 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | sniota 5233 | . . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))}) | |
3 | 1, 2 | sylbi 121 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))}) |
4 | df-rab 2477 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
5 | df-riota 5861 | . . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
6 | 5 | sneqi 3626 | . 2 ⊢ {(℩𝑥 ∈ 𝐴 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))} |
7 | 3, 4, 6 | 3eqtr4g 2247 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {(℩𝑥 ∈ 𝐴 𝜑)}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∃!weu 2038 ∈ wcel 2160 {cab 2175 ∃!wreu 2470 {crab 2472 {csn 3614 ℩cio 5201 ℩crio 5860 |
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-rex 2474 df-reu 2475 df-rab 2477 df-v 2758 df-sbc 2982 df-un 3153 df-sn 3620 df-pr 3621 df-uni 3832 df-iota 5203 df-riota 5861 |
This theorem is referenced by: divalgmod 12042 |
Copyright terms: Public domain | W3C validator |