Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dfnul3 | Structured version Visualization version 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 | pm3.24 405 | . . . . 5 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴) | |
2 | equid 2019 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
3 | 1, 2 | 2th 266 | . . . 4 ⊢ (¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴) ↔ 𝑥 = 𝑥) |
4 | 3 | con1bii 359 | . . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)) |
5 | 4 | abbii 2886 | . 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)} |
6 | dfnul2 4293 | . 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} | |
7 | df-rab 3147 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)} | |
8 | 5, 6, 7 | 3eqtr4i 2854 | 1 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2799 {crab 3142 ∅c0 4291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-rab 3147 df-dif 3939 df-nul 4292 |
This theorem is referenced by: difidALT 4331 kmlem3 9578 |
Copyright terms: Public domain | W3C validator |