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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 4357 | . . . . . . 7 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | ssel 3988 | . . . . . . . . . . 11 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵})) | |
3 | elsni 4647 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
4 | 2, 3 | syl6 35 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 = 𝐵)) |
5 | eleq1 2826 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
6 | 4, 5 | syl6 35 | . . . . . . . . 9 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))) |
7 | 6 | ibd 269 | . . . . . . . 8 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
8 | 7 | exlimdv 1930 | . . . . . . 7 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
9 | 1, 8 | biimtrid 242 | . . . . . 6 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐵 ∈ 𝐴)) |
10 | snssi 4812 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
11 | 9, 10 | syl6 35 | . . . . 5 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → {𝐵} ⊆ 𝐴)) |
12 | 11 | anc2li 555 | . . . 4 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))) |
13 | eqss 4010 | . . . 4 ⊢ (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)) | |
14 | 12, 13 | imbitrrdi 252 | . . 3 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐴 = {𝐵})) |
15 | 14 | orrd 863 | . 2 ⊢ (𝐴 ⊆ {𝐵} → (𝐴 = ∅ ∨ 𝐴 = {𝐵})) |
16 | 0ss 4405 | . . . 4 ⊢ ∅ ⊆ {𝐵} | |
17 | sseq1 4020 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵})) | |
18 | 16, 17 | mpbiri 258 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵}) |
19 | eqimss 4053 | . . 3 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵}) | |
20 | 18, 19 | jaoi 857 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵}) |
21 | 15, 20 | impbii 209 | 1 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ⊆ wss 3962 ∅c0 4338 {csn 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-dif 3965 df-ss 3979 df-nul 4339 df-sn 4631 |
This theorem is referenced by: eqsn 4833 snsssn 4845 pwsn 4904 frsn 5775 foconst 6835 fin1a2lem12 10448 fpwwe2lem12 10679 gsumval2 18711 0top 23005 minveclem4a 25477 uvtx01vtx 29428 snsssng 32541 pmtrcnelor 33093 0ringsubrg 33237 lvecdim0 33633 locfinref 33801 ordcmp 36429 bj-snmoore 37095 nlpineqsn 37390 uneqsn 44014 mosssn 48662 mosssn2 48664 mofsssn 48675 |
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