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Mirrors > Home > MPE Home > Th. List > sssn | Structured version Visualization version GIF version |
Description: The subsets of a singleton. (Contributed by NM, 24-Apr-2004.) |
Ref | Expression |
---|---|
sssn | ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neq0 3889 | . . . . . . 7 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | ssel 3562 | . . . . . . . . . . 11 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵})) | |
3 | elsni 4142 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
4 | 2, 3 | syl6 34 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 = 𝐵)) |
5 | eleq1 2676 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
6 | 4, 5 | syl6 34 | . . . . . . . . 9 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))) |
7 | 6 | ibd 257 | . . . . . . . 8 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
8 | 7 | exlimdv 1848 | . . . . . . 7 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
9 | 1, 8 | syl5bi 231 | . . . . . 6 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐵 ∈ 𝐴)) |
10 | snssi 4280 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
11 | 9, 10 | syl6 34 | . . . . 5 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → {𝐵} ⊆ 𝐴)) |
12 | 11 | anc2li 578 | . . . 4 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))) |
13 | eqss 3583 | . . . 4 ⊢ (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)) | |
14 | 12, 13 | syl6ibr 241 | . . 3 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐴 = {𝐵})) |
15 | 14 | orrd 392 | . 2 ⊢ (𝐴 ⊆ {𝐵} → (𝐴 = ∅ ∨ 𝐴 = {𝐵})) |
16 | 0ss 3924 | . . . 4 ⊢ ∅ ⊆ {𝐵} | |
17 | sseq1 3589 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵})) | |
18 | 16, 17 | mpbiri 247 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵}) |
19 | eqimss 3620 | . . 3 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵}) | |
20 | 18, 19 | jaoi 393 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵}) |
21 | 15, 20 | impbii 198 | 1 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ⊆ wss 3540 ∅c0 3874 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 |
This theorem is referenced by: eqsn 4301 eqsnOLD 4302 snsssn 4312 pwsn 4366 frsn 5112 foconst 6039 fin1a2lem12 9116 fpwwe2lem13 9343 gsumval2 17103 0top 20598 minveclem4a 23009 uvtx01vtx 26020 locfinref 29236 ordcmp 31616 uneqsn 37341 uvtxa01vtx0 40623 |
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