| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > dfv2 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the universal class (see df-v 3482). (Contributed by BJ, 30-Nov-2019.) |
| Ref | Expression |
|---|---|
| dfv2 | ⊢ V = {𝑥 ∣ ⊤} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-v 3482 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
| 2 | equid 2011 | . . . 4 ⊢ 𝑥 = 𝑥 | |
| 3 | 2 | bitru 1549 | . . 3 ⊢ (𝑥 = 𝑥 ↔ ⊤) |
| 4 | 3 | abbii 2809 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ ⊤} |
| 5 | 1, 4 | eqtri 2765 | 1 ⊢ V = {𝑥 ∣ ⊤} |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ⊤wtru 1541 {cab 2714 Vcvv 3480 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-v 3482 |
| This theorem is referenced by: vex 3484 abv 3492 vn0 4345 ab0orv 4383 bj-abv 36907 |
| Copyright terms: Public domain | W3C validator |