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Mirrors > Home > ILE Home > Th. List > sssnm | GIF version |
Description: The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.) |
Ref | Expression |
---|---|
sssnm | ⊢ (∃x x ∈ A → (A ⊆ {B} ↔ A = {B})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2933 | . . . . . . . . . 10 ⊢ (A ⊆ {B} → (x ∈ A → x ∈ {B})) | |
2 | elsni 3391 | . . . . . . . . . 10 ⊢ (x ∈ {B} → x = B) | |
3 | 1, 2 | syl6 29 | . . . . . . . . 9 ⊢ (A ⊆ {B} → (x ∈ A → x = B)) |
4 | eleq1 2097 | . . . . . . . . 9 ⊢ (x = B → (x ∈ A ↔ B ∈ A)) | |
5 | 3, 4 | syl6 29 | . . . . . . . 8 ⊢ (A ⊆ {B} → (x ∈ A → (x ∈ A ↔ B ∈ A))) |
6 | 5 | ibd 167 | . . . . . . 7 ⊢ (A ⊆ {B} → (x ∈ A → B ∈ A)) |
7 | 6 | exlimdv 1697 | . . . . . 6 ⊢ (A ⊆ {B} → (∃x x ∈ A → B ∈ A)) |
8 | snssi 3499 | . . . . . 6 ⊢ (B ∈ A → {B} ⊆ A) | |
9 | 7, 8 | syl6 29 | . . . . 5 ⊢ (A ⊆ {B} → (∃x x ∈ A → {B} ⊆ A)) |
10 | 9 | anc2li 312 | . . . 4 ⊢ (A ⊆ {B} → (∃x x ∈ A → (A ⊆ {B} ∧ {B} ⊆ A))) |
11 | eqss 2954 | . . . 4 ⊢ (A = {B} ↔ (A ⊆ {B} ∧ {B} ⊆ A)) | |
12 | 10, 11 | syl6ibr 151 | . . 3 ⊢ (A ⊆ {B} → (∃x x ∈ A → A = {B})) |
13 | 12 | com12 27 | . 2 ⊢ (∃x x ∈ A → (A ⊆ {B} → A = {B})) |
14 | eqimss 2991 | . 2 ⊢ (A = {B} → A ⊆ {B}) | |
15 | 13, 14 | impbid1 130 | 1 ⊢ (∃x x ∈ A → (A ⊆ {B} ↔ A = {B})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ⊆ wss 2911 {csn 3367 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 df-sn 3373 |
This theorem is referenced by: eqsnm 3517 |
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