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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 3407 | . 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} | |
2 | ab0 4335 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = ∅ ↔ ∀𝑥 ¬ (𝑥 ∈ ∅ ∧ 𝜑)) | |
3 | noel 4291 | . . . 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 3406 ∅c0 4283 |
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 3407 df-dif 3914 df-nul 4284 |
This theorem is referenced by: rabsnif 4685 fvmptrabfv 6980 supp0 8098 sup00 9405 scott0 9827 psgnfval 19287 pmtrsn 19306 00lsp 20457 rrgval 20773 leftval 27215 rightval 27216 uvtx0 28384 vtxdg0e 28464 wwlksn 28824 wspthsn 28835 iswwlksnon 28840 iswspthsnon 28843 clwwlk0on0 29078 zar0ring 32516 satf0 34023 fvmptrab 45610 fvmptrabdm 45611 prprspr2 45796 |
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