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Mirrors > Home > ILE Home > Th. List > sseq0 | GIF version |
Description: A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
sseq0 | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 3021 | . . 3 ⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ ∅)) | |
2 | ss0 3284 | . . 3 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
3 | 1, 2 | syl6bi 161 | . 2 ⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 → 𝐴 = ∅)) |
4 | 3 | impcom 123 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ⊆ wss 2973 ∅c0 3251 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-in 2979 df-ss 2986 df-nul 3252 |
This theorem is referenced by: ssn0 3286 ssdifin0 3324 |
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