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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 3434 | . 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} | |
2 | ab0 4375 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ ↔ ∀𝑥 ¬ (𝑥 ∈ ∅ ∧ 𝜑)) | |
3 | noel 4331 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
4 | 3 | intnanr 489 | . . 3 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑) |
5 | 2, 4 | mpgbir 1802 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ |
6 | 1, 5 | eqtri 2761 | 1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 {crab 3433 ∅c0 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-dif 3952 df-nul 4324 |
This theorem is referenced by: rabsnif 4728 fvmptrabfv 7030 supp0 8151 sup00 9459 scott0 9881 psgnfval 19368 pmtrsn 19387 00lsp 20592 rrgval 20903 leftval 27358 rightval 27359 uvtx0 28651 vtxdg0e 28731 wwlksn 29091 wspthsn 29102 iswwlksnon 29107 iswspthsnon 29110 clwwlk0on0 29345 zar0ring 32858 satf0 34363 fvmptrab 46000 fvmptrabdm 46001 prprspr2 46186 |
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