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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 3408 | . . . . 5 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | intnanr 920 | . . . 4 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑) |
3 | equid 1688 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
4 | 3 | notnoti 635 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
5 | 2, 4 | 2false 691 | . . 3 ⊢ ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥) |
6 | 5 | abbii 2280 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
7 | df-rab 2451 | . 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} | |
8 | dfnul2 3406 | . 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} | |
9 | 6, 7, 8 | 3eqtr4i 2195 | 1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1342 ∈ wcel 2135 {cab 2150 {crab 2446 ∅c0 3404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rab 2451 df-v 2723 df-dif 3113 df-nul 3405 |
This theorem is referenced by: ssfirab 6890 sup00 6959 |
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