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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 3164 | . . . . . . . . . 10 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵})) | |
2 | elsni 3625 | . . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵) | |
3 | 1, 2 | syl6 33 | . . . . . . . . 9 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 = 𝐵)) |
4 | eleq1 2252 | . . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
5 | 3, 4 | syl6 33 | . . . . . . . 8 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))) |
6 | 5 | ibd 178 | . . . . . . 7 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
7 | 6 | exlimdv 1830 | . . . . . 6 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
8 | snssi 3751 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
9 | 7, 8 | syl6 33 | . . . . 5 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → {𝐵} ⊆ 𝐴)) |
10 | 9 | anc2li 329 | . . . 4 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))) |
11 | eqss 3185 | . . . 4 ⊢ (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)) | |
12 | 10, 11 | imbitrrdi 162 | . . 3 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 = {𝐵})) |
13 | 12 | com12 30 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} → 𝐴 = {𝐵})) |
14 | eqimss 3224 | . 2 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵}) | |
15 | 13, 14 | impbid1 142 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∃wex 1503 ∈ wcel 2160 ⊆ wss 3144 {csn 3607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-ss 3157 df-sn 3613 |
This theorem is referenced by: eqsnm 3770 ss1o0el1 4215 exmidn0m 4219 exmidsssn 4220 exmidomni 7171 exmidunben 12480 exmidsbthrlem 15249 sbthomlem 15252 |
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