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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 4246 | . . . . . . 7 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | ssel 3880 | . . . . . . . . . . 11 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵})) | |
3 | elsni 4544 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
4 | 2, 3 | syl6 35 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 = 𝐵)) |
5 | eleq1 2818 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
6 | 4, 5 | syl6 35 | . . . . . . . . 9 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))) |
7 | 6 | ibd 272 | . . . . . . . 8 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
8 | 7 | exlimdv 1941 | . . . . . . 7 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
9 | 1, 8 | syl5bi 245 | . . . . . 6 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐵 ∈ 𝐴)) |
10 | snssi 4707 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
11 | 9, 10 | syl6 35 | . . . . 5 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → {𝐵} ⊆ 𝐴)) |
12 | 11 | anc2li 559 | . . . 4 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))) |
13 | eqss 3902 | . . . 4 ⊢ (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)) | |
14 | 12, 13 | syl6ibr 255 | . . 3 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐴 = {𝐵})) |
15 | 14 | orrd 863 | . 2 ⊢ (𝐴 ⊆ {𝐵} → (𝐴 = ∅ ∨ 𝐴 = {𝐵})) |
16 | 0ss 4297 | . . . 4 ⊢ ∅ ⊆ {𝐵} | |
17 | sseq1 3912 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵})) | |
18 | 16, 17 | mpbiri 261 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵}) |
19 | eqimss 3943 | . . 3 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵}) | |
20 | 18, 19 | jaoi 857 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵}) |
21 | 15, 20 | impbii 212 | 1 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ⊆ wss 3853 ∅c0 4223 {csn 4527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-dif 3856 df-in 3860 df-ss 3870 df-nul 4224 df-sn 4528 |
This theorem is referenced by: eqsn 4728 snsssn 4738 pwsn 4797 frsn 5621 foconst 6626 fin1a2lem12 9990 fpwwe2lem12 10221 gsumval2 18112 0top 21834 minveclem4a 24281 uvtx01vtx 27439 snsssng 30533 pmtrcnelor 31033 lvecdim0 31358 locfinref 31459 ordcmp 34322 bj-snmoore 34968 nlpineqsn 35265 uneqsn 41251 mosssn 45776 mosssn2 45778 mofsssn 45789 |
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