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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 3433 | . 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} | |
2 | ab0 4373 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ ↔ ∀𝑥 ¬ (𝑥 ∈ ∅ ∧ 𝜑)) | |
3 | noel 4329 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
4 | 3 | intnanr 488 | . . 3 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑) |
5 | 2, 4 | mpgbir 1801 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ |
6 | 1, 5 | eqtri 2760 | 1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2709 {crab 3432 ∅c0 4321 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-dif 3950 df-nul 4322 |
This theorem is referenced by: rabsnif 4726 fvmptrabfv 7026 supp0 8147 sup00 9455 scott0 9877 psgnfval 19362 pmtrsn 19381 00lsp 20584 rrgval 20895 leftval 27347 rightval 27348 uvtx0 28640 vtxdg0e 28720 wwlksn 29080 wspthsn 29091 iswwlksnon 29096 iswspthsnon 29099 clwwlk0on0 29334 zar0ring 32846 satf0 34351 fvmptrab 45986 fvmptrabdm 45987 prprspr2 46172 |
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