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Mirrors > Home > ILE Home > Th. List > Mathboxes > sssneq | GIF version |
Description: Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.) |
Ref | Expression |
---|---|
sssneq | ⊢ (𝐴 ⊆ {𝐵} → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . 5 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐴 ⊆ {𝐵}) | |
2 | simprl 521 | . . . . 5 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 ∈ 𝐴) | |
3 | 1, 2 | sseldd 3125 | . . . 4 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 ∈ {𝐵}) |
4 | elsni 3574 | . . . 4 ⊢ (𝑦 ∈ {𝐵} → 𝑦 = 𝐵) | |
5 | 3, 4 | syl 14 | . . 3 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 = 𝐵) |
6 | simprr 522 | . . . . 5 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝐴) | |
7 | 1, 6 | sseldd 3125 | . . . 4 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ {𝐵}) |
8 | elsni 3574 | . . . 4 ⊢ (𝑧 ∈ {𝐵} → 𝑧 = 𝐵) | |
9 | 7, 8 | syl 14 | . . 3 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 = 𝐵) |
10 | 5, 9 | eqtr4d 2190 | . 2 ⊢ ((𝐴 ⊆ {𝐵} ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑦 = 𝑧) |
11 | 10 | ralrimivva 2536 | 1 ⊢ (𝐴 ⊆ {𝐵} → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑦 = 𝑧) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 2125 ∀wral 2432 ⊆ wss 3098 {csn 3556 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-ext 2136 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1740 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ral 2437 df-v 2711 df-in 3104 df-ss 3111 df-sn 3562 |
This theorem is referenced by: (None) |
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