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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 4719 | . 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) | |
2 | sneqr.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
3 | 2 | snnz 4672 | . . . . 5 ⊢ {𝐴} ≠ ∅ |
4 | 3 | neii 2989 | . . . 4 ⊢ ¬ {𝐴} = ∅ |
5 | 4 | pm2.21i 119 | . . 3 ⊢ ({𝐴} = ∅ → 𝐴 = 𝐵) |
6 | 2 | sneqr 4731 | . . 3 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵) |
7 | 5, 6 | jaoi 854 | . 2 ⊢ (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵) |
8 | 1, 7 | sylbi 220 | 1 ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 |
This theorem is referenced by: k0004lem3 40852 |
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