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Mirrors > Home > ILE Home > Th. List > rab0 | GIF version |
Description: Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
rab0 | ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3255 | . . . . 5 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | intnanr 872 | . . . 4 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑) |
3 | equid 1629 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
4 | 3 | notnoti 606 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
5 | 2, 4 | 2false 649 | . . 3 ⊢ ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥) |
6 | 5 | abbii 2194 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
7 | df-rab 2357 | . 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} | |
8 | dfnul2 3253 | . 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} | |
9 | 6, 7, 8 | 3eqtr4i 2111 | 1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 102 = wceq 1284 ∈ wcel 1433 {cab 2067 {crab 2352 ∅c0 3251 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rab 2357 df-v 2603 df-dif 2975 df-nul 3252 |
This theorem is referenced by: sup00 6416 |
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