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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 3115 | . 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} | |
2 | ab0 4287 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ ↔ ∀𝑥 ¬ (𝑥 ∈ ∅ ∧ 𝜑)) | |
3 | noel 4247 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
4 | 3 | intnanr 491 | . . 3 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑) |
5 | 2, 4 | mpgbir 1801 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ |
6 | 1, 5 | eqtri 2821 | 1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 {crab 3110 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-dif 3884 df-nul 4244 |
This theorem is referenced by: rabsnif 4619 fvmptrabfv 6776 supp0 7818 sup00 8912 scott0 9299 psgnfval 18620 pmtrsn 18639 00lsp 19746 rrgval 20053 uvtx0 27184 vtxdg0e 27264 wwlksn 27623 wspthsn 27634 iswwlksnon 27639 iswspthsnon 27642 clwwlk0on0 27877 zar0ring 31231 satf0 32732 fvmptrab 43848 fvmptrabdm 43849 prprspr2 44035 |
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