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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 4298 | . 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) | |
2 | sneqr.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
3 | 2 | snnz 4252 | . . . . 5 ⊢ {𝐴} ≠ ∅ |
4 | 3 | neii 2784 | . . . 4 ⊢ ¬ {𝐴} = ∅ |
5 | 4 | pm2.21i 115 | . . 3 ⊢ ({𝐴} = ∅ → 𝐴 = 𝐵) |
6 | 2 | sneqr 4311 | . . 3 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵) |
7 | 5, 6 | jaoi 393 | . 2 ⊢ (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵) |
8 | 1, 7 | sylbi 206 | 1 ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 |
This theorem is referenced by: k0004lem3 37467 |
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