Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > euabsn2 | Structured version Visualization version 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 2462 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
2 | abeq1 2720 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ {𝑦})) | |
3 | velsn 4141 | . . . . . 6 ⊢ (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦) | |
4 | 3 | bibi2i 326 | . . . . 5 ⊢ ((𝜑 ↔ 𝑥 ∈ {𝑦}) ↔ (𝜑 ↔ 𝑥 = 𝑦)) |
5 | 4 | albii 1737 | . . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
6 | 2, 5 | bitri 263 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
7 | 6 | exbii 1764 | . 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
8 | 1, 7 | bitr4i 266 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃!weu 2458 {cab 2596 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sn 4126 |
This theorem is referenced by: euabsn 4205 reusn 4206 absneu 4207 uniintab 4450 eusvobj2 6542 |
Copyright terms: Public domain | W3C validator |