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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 4762 | . 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) | |
2 | sneqr.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
3 | 2 | snnz 4714 | . . . . 5 ⊢ {𝐴} ≠ ∅ |
4 | 3 | neii 3021 | . . . 4 ⊢ ¬ {𝐴} = ∅ |
5 | 4 | pm2.21i 119 | . . 3 ⊢ ({𝐴} = ∅ → 𝐴 = 𝐵) |
6 | 2 | sneqr 4774 | . . 3 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵) |
7 | 5, 6 | jaoi 853 | . 2 ⊢ (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵) |
8 | 1, 7 | sylbi 219 | 1 ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 ∅c0 4294 {csn 4570 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-v 3499 df-dif 3942 df-in 3946 df-ss 3955 df-nul 4295 df-sn 4571 |
This theorem is referenced by: k0004lem3 40505 |
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