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Mirrors > Home > MPE Home > Th. List > rab0 | Structured version Visualization version 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.) (Proof shortened by JJ, 14-Jul-2021.) |
Ref | Expression |
---|---|
rab0 | ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2905 | . 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} | |
2 | ab0 3905 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ ↔ ∀𝑥 ¬ (𝑥 ∈ ∅ ∧ 𝜑)) | |
3 | noel 3878 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
4 | 3 | intnanr 952 | . . 3 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑) |
5 | 2, 4 | mpgbir 1717 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ |
6 | 1, 5 | eqtri 2632 | 1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 {crab 2900 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-nul 3875 |
This theorem is referenced by: rabsnif 4202 supp0 7187 sup00 8253 scott0 8632 psgnfval 17743 pmtrsn 17762 00lsp 18802 rrgval 19108 usgra0v 25900 vdgr0 26427 uvtxa0 40620 vtxdg0e 40689 wwlksn 41040 wspthsn 41046 iswwlksnon 41051 iswspthsnon 41052 clwwlksn 41189 |
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