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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 636 | . . 3 ⊢ (𝜑 → 𝑥 ∈ ∅) |
3 | 2 | abssi 3177 | . 2 ⊢ {𝑥 ∣ 𝜑} ⊆ ∅ |
4 | ss0 3408 | . 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 1332 ∈ wcel 1481 {cab 2126 ⊆ wss 3076 ∅c0 3368 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-dif 3078 df-in 3082 df-ss 3089 df-nul 3369 |
This theorem is referenced by: csbprc 3413 mpo0 5849 fi0 6871 |
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