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Mirrors > Home > ILE Home > Th. List > ss0b | GIF version |
Description: Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.) |
Ref | Expression |
---|---|
ss0b | ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3371 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | eqss 3082 | . . 3 ⊢ (𝐴 = ∅ ↔ (𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴)) | |
3 | 1, 2 | mpbiran2 910 | . 2 ⊢ (𝐴 = ∅ ↔ 𝐴 ⊆ ∅) |
4 | 3 | bicomi 131 | 1 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 ⊆ wss 3041 ∅c0 3333 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-dif 3043 df-in 3047 df-ss 3054 df-nul 3334 |
This theorem is referenced by: ss0 3373 un00 3379 ssdisj 3389 pw0 3637 card0 7012 0nnei 12249 |
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