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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 2397 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | sniota 5073 | . . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))}) | |
3 | 1, 2 | sylbi 120 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))}) |
4 | df-rab 2399 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
5 | df-riota 5684 | . . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
6 | 5 | sneqi 3505 | . 2 ⊢ {(℩𝑥 ∈ 𝐴 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))} |
7 | 3, 4, 6 | 3eqtr4g 2172 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {(℩𝑥 ∈ 𝐴 𝜑)}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1314 ∈ wcel 1463 ∃!weu 1975 {cab 2101 ∃!wreu 2392 {crab 2394 {csn 3493 ℩cio 5044 ℩crio 5683 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-un 3041 df-sn 3499 df-pr 3500 df-uni 3703 df-iota 5046 df-riota 5684 |
This theorem is referenced by: divalgmod 11472 |
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