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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 3640 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
2 | nfv 1515 | . . 3 ⊢ Ⅎ𝑦{𝑥 ∣ 𝜑} = {𝑥} | |
3 | nfab1 2308 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
4 | 3 | nfeq1 2316 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} = {𝑦} |
5 | sneq 3582 | . . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
6 | 5 | eqeq2d 2176 | . . 3 ⊢ (𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑥} ↔ {𝑥 ∣ 𝜑} = {𝑦})) |
7 | 2, 4, 6 | cbvex 1743 | . 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 1342 ∃wex 1479 ∃!weu 2013 {cab 2150 {csn 3571 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2724 df-sn 3577 |
This theorem is referenced by: eusn 3645 args 4968 mapsn 6648 |
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