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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 3930 | . . . . . . 7 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | ssel 3597 | . . . . . . . . . . 11 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵})) | |
3 | elsni 4194 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
4 | 2, 3 | syl6 35 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 = 𝐵)) |
5 | eleq1 2689 | . . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
6 | 4, 5 | syl6 35 | . . . . . . . . 9 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))) |
7 | 6 | ibd 258 | . . . . . . . 8 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
8 | 7 | exlimdv 1861 | . . . . . . 7 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
9 | 1, 8 | syl5bi 232 | . . . . . 6 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐵 ∈ 𝐴)) |
10 | snssi 4339 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
11 | 9, 10 | syl6 35 | . . . . 5 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → {𝐵} ⊆ 𝐴)) |
12 | 11 | anc2li 580 | . . . 4 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))) |
13 | eqss 3618 | . . . 4 ⊢ (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)) | |
14 | 12, 13 | syl6ibr 242 | . . 3 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐴 = {𝐵})) |
15 | 14 | orrd 393 | . 2 ⊢ (𝐴 ⊆ {𝐵} → (𝐴 = ∅ ∨ 𝐴 = {𝐵})) |
16 | 0ss 3972 | . . . 4 ⊢ ∅ ⊆ {𝐵} | |
17 | sseq1 3626 | . . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵})) | |
18 | 16, 17 | mpbiri 248 | . . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵}) |
19 | eqimss 3657 | . . 3 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵}) | |
20 | 18, 19 | jaoi 394 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵}) |
21 | 15, 20 | impbii 199 | 1 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ⊆ wss 3574 ∅c0 3915 {csn 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 |
This theorem is referenced by: eqsn 4361 eqsnOLD 4362 snsssn 4372 pwsn 4428 frsn 5189 foconst 6126 fin1a2lem12 9233 fpwwe2lem13 9464 gsumval2 17280 0top 20787 minveclem4a 23201 uvtxa01vtx0 26297 locfinref 29908 ordcmp 32446 bj-snmoore 33068 uneqsn 38321 |
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