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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 3256 | . . . . 5 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | intnanr 873 | . . . 4 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑) |
3 | equid 1630 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
4 | 3 | notnoti 607 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
5 | 2, 4 | 2false 650 | . . 3 ⊢ ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥) |
6 | 5 | abbii 2195 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
7 | df-rab 2358 | . 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} | |
8 | dfnul2 3254 | . 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} | |
9 | 6, 7, 8 | 3eqtr4i 2112 | 1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 102 = wceq 1285 ∈ wcel 1434 {cab 2068 {crab 2353 ∅c0 3252 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-rab 2358 df-v 2604 df-dif 2976 df-nul 3253 |
This theorem is referenced by: sup00 6465 |
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