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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 3004 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵})) | |
2 | elsni 3440 | . . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
3 | 1, 2 | syl6 33 | . . . . . . . . 9 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 = 𝐵)) |
4 | eleq1 2145 | . . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
5 | 3, 4 | syl6 33 | . . . . . . . 8 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))) |
6 | 5 | ibd 176 | . . . . . . 7 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
7 | 6 | exlimdv 1742 | . . . . . 6 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
8 | snssi 3555 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
9 | 7, 8 | syl6 33 | . . . . 5 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → {𝐵} ⊆ 𝐴)) |
10 | 9 | anc2li 322 | . . . 4 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))) |
11 | eqss 3025 | . . . 4 ⊢ (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)) | |
12 | 10, 11 | syl6ibr 160 | . . 3 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 = {𝐵})) |
13 | 12 | com12 30 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} → 𝐴 = {𝐵})) |
14 | eqimss 3062 | . 2 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵}) | |
15 | 13, 14 | impbid1 140 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∃wex 1422 ∈ wcel 1434 ⊆ wss 2984 {csn 3422 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 df-in 2990 df-ss 2997 df-sn 3428 |
This theorem is referenced by: eqsnm 3573 exmid01 3995 exmidomni 6701 |
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