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Mirrors > Home > ILE Home > Th. List > euabsn | GIF version |
Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.) |
Ref | Expression |
---|---|
euabsn | ⊢ (∃!𝑥𝜑 ↔ ∃𝑥{𝑥 ∣ 𝜑} = {𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euabsn2 3592 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
2 | nfv 1508 | . . 3 ⊢ Ⅎ𝑦{𝑥 ∣ 𝜑} = {𝑥} | |
3 | nfab1 2283 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
4 | 3 | nfeq1 2291 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} = {𝑦} |
5 | sneq 3538 | . . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
6 | 5 | eqeq2d 2151 | . . 3 ⊢ (𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑥} ↔ {𝑥 ∣ 𝜑} = {𝑦})) |
7 | 2, 4, 6 | cbvex 1729 | . 2 ⊢ (∃𝑥{𝑥 ∣ 𝜑} = {𝑥} ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
8 | 1, 7 | bitr4i 186 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥{𝑥 ∣ 𝜑} = {𝑥}) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∃wex 1468 ∃!weu 1999 {cab 2125 {csn 3527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-sn 3533 |
This theorem is referenced by: eusn 3597 args 4908 mapsn 6584 |
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