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Mirrors > Home > ILE Home > Th. List > rab0 | 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.) |
Ref | Expression |
---|---|
rab0 | ⊢ {x ∈ ∅ ∣ φ} = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3222 | . . . . 5 ⊢ ¬ x ∈ ∅ | |
2 | 1 | intnanr 838 | . . . 4 ⊢ ¬ (x ∈ ∅ ∧ φ) |
3 | equid 1586 | . . . . 5 ⊢ x = x | |
4 | 3 | notnoti 573 | . . . 4 ⊢ ¬ ¬ x = x |
5 | 2, 4 | 2false 616 | . . 3 ⊢ ((x ∈ ∅ ∧ φ) ↔ ¬ x = x) |
6 | 5 | abbii 2150 | . 2 ⊢ {x ∣ (x ∈ ∅ ∧ φ)} = {x ∣ ¬ x = x} |
7 | df-rab 2309 | . 2 ⊢ {x ∈ ∅ ∣ φ} = {x ∣ (x ∈ ∅ ∧ φ)} | |
8 | dfnul2 3220 | . 2 ⊢ ∅ = {x ∣ ¬ x = x} | |
9 | 6, 7, 8 | 3eqtr4i 2067 | 1 ⊢ {x ∈ ∅ ∣ φ} = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 = wceq 1242 ∈ wcel 1390 {cab 2023 {crab 2304 ∅c0 3218 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rab 2309 df-v 2553 df-dif 2914 df-nul 3219 |
This theorem is referenced by: (None) |
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