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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 3476 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | eqss 3185 | . . 3 ⊢ (𝐴 = ∅ ↔ (𝐴 ⊆ ∅ ∧ ∅ ⊆ 𝐴)) | |
3 | 1, 2 | mpbiran2 943 | . 2 ⊢ (𝐴 = ∅ ↔ 𝐴 ⊆ ∅) |
4 | 3 | bicomi 132 | 1 ⊢ (𝐴 ⊆ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 ⊆ wss 3144 ∅c0 3437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-in 3150 df-ss 3157 df-nul 3438 |
This theorem is referenced by: ss0 3478 un00 3484 ssdisj 3494 pw0 3754 card0 7216 0nnei 14105 |
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