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Mirrors > Home > ILE Home > Th. List > snsssn | GIF version |
Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.) |
Ref | Expression |
---|---|
sneqr.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
snsssn | ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 2988 | . . 3 ⊢ ({𝐴} ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵})) | |
2 | velsn 3415 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
3 | velsn 3415 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
4 | 2, 3 | imbi12i 237 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴 → 𝑥 = 𝐵)) |
5 | 4 | albii 1399 | . . 3 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵)) |
6 | 1, 5 | bitri 182 | . 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵)) |
7 | sneqr.1 | . . 3 ⊢ 𝐴 ∈ V | |
8 | sbceqal 2869 | . . 3 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵)) | |
9 | 7, 8 | ax-mp 7 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵) |
10 | 6, 9 | sylbi 119 | 1 ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ⊆ wss 2973 {csn 3398 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 df-in 2979 df-ss 2986 df-sn 3404 |
This theorem is referenced by: (None) |
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