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| Mirrors > Home > MPE Home > Th. List > snsssn | 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.) |
| Ref | Expression |
|---|---|
| sneqr.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| snsssn | ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sssn 4826 | . 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) | |
| 2 | sneqr.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
| 3 | 2 | snnz 4776 | . . . . 5 ⊢ {𝐴} ≠ ∅ |
| 4 | 3 | neii 2942 | . . . 4 ⊢ ¬ {𝐴} = ∅ |
| 5 | 4 | pm2.21i 119 | . . 3 ⊢ ({𝐴} = ∅ → 𝐴 = 𝐵) |
| 6 | 2 | sneqr 4840 | . . 3 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵) |
| 7 | 5, 6 | jaoi 858 | . 2 ⊢ (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵) |
| 8 | 1, 7 | sylbi 217 | 1 ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ 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: k0004lem3 44162 |
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