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Mirrors > Home > ILE Home > Th. List > euabsn2 | GIF version |
Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
euabsn2 | ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 2002 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
2 | abeq1 2249 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ {𝑦})) | |
3 | velsn 3544 | . . . . . 6 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦) | |
4 | 3 | bibi2i 226 | . . . . 5 ⊢ ((𝜑 ↔ 𝑥 ∈ {𝑦}) ↔ (𝜑 ↔ 𝑥 = 𝑦)) |
5 | 4 | albii 1446 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
6 | 2, 5 | bitri 183 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
7 | 6 | exbii 1584 | . 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
8 | 1, 7 | bitr4i 186 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wal 1329 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∃!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: euabsn 3593 reusn 3594 absneu 3595 uniintabim 3808 euabex 4147 nfvres 5454 eusvobj2 5760 |
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