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| Mirrors > Home > MPE Home > Th. List > Mathboxes > snsssng | Structured version Visualization version GIF version | ||
| Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.) (Revised by Thierry Arnoux, 11-Apr-2024.) | 
| Ref | Expression | 
|---|---|
| snsssng | ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sssn 4826 | . 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) | |
| 2 | snnzg 4774 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅) | |
| 3 | 2 | neneqd 2945 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} = ∅) | 
| 4 | 3 | pm2.21d 121 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = ∅ → 𝐴 = 𝐵)) | 
| 5 | sneqrg 4839 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) | |
| 6 | 4, 5 | jaod 860 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵)) | 
| 7 | 6 | imp 406 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) → 𝐴 = 𝐵) | 
| 8 | 1, 7 | sylan2b 594 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∅c0 4333 {csn 4626 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-sn 4627 | 
| This theorem is referenced by: (None) | 
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