Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > abf | GIF version |
Description: A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.) |
Ref | Expression |
---|---|
abf.1 | ⊢ ¬ 𝜑 |
Ref | Expression |
---|---|
abf | ⊢ {𝑥 ∣ 𝜑} = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abf.1 | . . . 4 ⊢ ¬ 𝜑 | |
2 | 1 | pm2.21i 635 | . . 3 ⊢ (𝜑 → 𝑥 ∈ ∅) |
3 | 2 | abssi 3167 | . 2 ⊢ {𝑥 ∣ 𝜑} ⊆ ∅ |
4 | ss0 3398 | . 2 ⊢ ({𝑥 ∣ 𝜑} ⊆ ∅ → {𝑥 ∣ 𝜑} = ∅) | |
5 | 3, 4 | ax-mp 5 | 1 ⊢ {𝑥 ∣ 𝜑} = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1331 ∈ wcel 1480 {cab 2123 ⊆ wss 3066 ∅c0 3358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-dif 3068 df-in 3072 df-ss 3079 df-nul 3359 |
This theorem is referenced by: csbprc 3403 mpo0 5834 fi0 6856 |
Copyright terms: Public domain | W3C validator |