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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 4055. (Contributed by Jim Kingdon, 10-Aug-2018.) |
Ref | Expression |
---|---|
sssnr | ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3340 | . . 3 ⊢ ∅ ⊆ {𝐵} | |
2 | sseq1 3062 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵})) | |
3 | 1, 2 | mpbiri 167 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵}) |
4 | eqimss 3093 | . 2 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵}) | |
5 | 3, 4 | jaoi 674 | 1 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 667 = wceq 1296 ⊆ wss 3013 ∅c0 3302 {csn 3466 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-dif 3015 df-in 3019 df-ss 3026 df-nul 3303 |
This theorem is referenced by: pwsnss 3669 |
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