![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3333 | . . . . 5 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | intnanr 898 | . . . 4 ⊢ ¬ (𝑥 ∈ ∅ ∧ 𝜑) |
3 | equid 1660 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
4 | 3 | notnoti 617 | . . . 4 ⊢ ¬ ¬ 𝑥 = 𝑥 |
5 | 2, 4 | 2false 673 | . . 3 ⊢ ((𝑥 ∈ ∅ ∧ 𝜑) ↔ ¬ 𝑥 = 𝑥) |
6 | 5 | abbii 2230 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} = {𝑥 ∣ ¬ 𝑥 = 𝑥} |
7 | df-rab 2399 | . 2 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ ∅ ∧ 𝜑)} | |
8 | dfnul2 3331 | . 2 ⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} | |
9 | 6, 7, 8 | 3eqtr4i 2145 | 1 ⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 = wceq 1314 ∈ wcel 1463 {cab 2101 {crab 2394 ∅c0 3329 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-rab 2399 df-v 2659 df-dif 3039 df-nul 3330 |
This theorem is referenced by: ssfirab 6774 sup00 6842 |
Copyright terms: Public domain | W3C validator |