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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 3228 | . . . . 5 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | intnanr 839 | . . . 4 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑) |
3 | equid 1589 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
4 | 3 | notnoti 574 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
5 | 2, 4 | 2false 617 | . . 3 ⊢ ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥) |
6 | 5 | abbii 2153 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
7 | df-rab 2315 | . 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} | |
8 | dfnul2 3226 | . 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} | |
9 | 6, 7, 8 | 3eqtr4i 2070 | 1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ |
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: (None) |
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