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Mirrors > Home > ILE Home > Th. List > sssnr | GIF version |
Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4214. (Contributed by Jim Kingdon, 10-Aug-2018.) |
Ref | Expression |
---|---|
sssnr | ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3473 | . . 3 ⊢ ∅ ⊆ {𝐵} | |
2 | sseq1 3190 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵})) | |
3 | 1, 2 | mpbiri 168 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵}) |
4 | eqimss 3221 | . 2 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵}) | |
5 | 3, 4 | jaoi 717 | 1 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 709 = wceq 1363 ⊆ wss 3141 ∅c0 3434 {csn 3604 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-dif 3143 df-in 3147 df-ss 3154 df-nul 3435 |
This theorem is referenced by: pwsnss 3815 |
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