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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 3081 | . . 3 ⊢ ({𝐴} ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵})) | |
2 | velsn 3539 | . . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
3 | velsn 3539 | . . . . 5 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
4 | 2, 3 | imbi12i 238 | . . . 4 ⊢ ((𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}) ↔ (𝑥 = 𝐴 → 𝑥 = 𝐵)) |
5 | 4 | albii 1446 | . . 3 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵)) |
6 | 1, 5 | bitri 183 | . 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵)) |
7 | sneqr.1 | . . 3 ⊢ 𝐴 ∈ V | |
8 | sbceqal 2959 | . . 3 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵)) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵) |
10 | 6, 9 | sylbi 120 | 1 ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 = wceq 1331 ∈ wcel 1480 Vcvv 2681 ⊆ wss 3066 {csn 3522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-sbc 2905 df-in 3072 df-ss 3079 df-sn 3528 |
This theorem is referenced by: (None) |
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