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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 1701 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | 1 | notnoti 645 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
3 | pm3.24 693 | . . . 4 ⊢ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴) | |
4 | 2, 3 | 2false 701 | . . 3 ⊢ (¬ 𝑥 = 𝑥 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)) |
5 | 4 | abbii 2293 | . 2 ⊢ {𝑥 ∣ ¬ 𝑥 = 𝑥} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)} |
6 | dfnul2 3424 | . 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} | |
7 | df-rab 2464 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐴)} | |
8 | 5, 6, 7 | 3eqtr4i 2208 | 1 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 = wceq 1353 ∈ wcel 2148 {cab 2163 {crab 2459 ∅c0 3422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rab 2464 df-v 2739 df-dif 3131 df-nul 3423 |
This theorem is referenced by: difidALT 3492 |
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