Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dfnul3 | GIF version |
Description: Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) |
Ref | Expression |
---|---|
dfnul3 | ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 1589 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | 1 | notnoti 574 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
3 | pm3.24 627 | . . . 4 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴) | |
4 | 2, 3 | 2false 617 | . . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)) |
5 | 4 | abbii 2153 | . 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)} |
6 | dfnul2 3226 | . 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} | |
7 | df-rab 2315 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)} | |
8 | 5, 6, 7 | 3eqtr4i 2070 | 1 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 = wceq 1243 ∈ wcel 1393 {cab 2026 {crab 2310 ∅c0 3224 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rab 2315 df-v 2559 df-dif 2920 df-nul 3225 |
This theorem is referenced by: difidALT 3293 |
Copyright terms: Public domain | W3C validator |