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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 3033 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵})) | |
2 | elsni 3484 | . . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
3 | 1, 2 | syl6 33 | . . . . . . . . 9 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 = 𝐵)) |
4 | eleq1 2157 | . . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
5 | 3, 4 | syl6 33 | . . . . . . . 8 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))) |
6 | 5 | ibd 177 | . . . . . . 7 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
7 | 6 | exlimdv 1754 | . . . . . 6 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
8 | snssi 3603 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
9 | 7, 8 | syl6 33 | . . . . 5 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → {𝐵} ⊆ 𝐴)) |
10 | 9 | anc2li 323 | . . . 4 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))) |
11 | eqss 3054 | . . . 4 ⊢ (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)) | |
12 | 10, 11 | syl6ibr 161 | . . 3 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 = {𝐵})) |
13 | 12 | com12 30 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} → 𝐴 = {𝐵})) |
14 | eqimss 3093 | . 2 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵}) | |
15 | 13, 14 | impbid1 141 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1296 ∃wex 1433 ∈ wcel 1445 ⊆ wss 3013 {csn 3466 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-in 3019 df-ss 3026 df-sn 3472 |
This theorem is referenced by: eqsnm 3621 exmid01 4053 exmidn0m 4054 exmidsssn 4055 exmidomni 6885 exmidsbthrlem 12633 |
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