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Mirrors > Home > MPE Home > Th. List > elsn2 | Structured version Visualization version GIF version |
Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that 𝐵, rather than 𝐴, be a set. (Contributed by NM, 12-Jun-1994.) |
Ref | Expression |
---|---|
elsn2.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
elsn2 | ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsn2.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | elsn2g 4629 | . 2 ⊢ (𝐵 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3448 {csn 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-sn 4592 |
This theorem is referenced by: fparlem1 8049 fparlem2 8050 el1o 8446 fin1a2lem11 10353 fin1a2lem12 10354 elnn0 12422 elxnn0 12494 elfzp1 13498 fsumss 15617 fprodss 15838 elhoma 17925 islpidl 20732 zrhrhmb 20927 rest0 22536 qustgphaus 23490 taylfval 25734 elch0 30238 atoml2i 31367 prmidl0 32263 bj-eltag 35477 bj-rest10b 35589 dibopelvalN 39635 dibopelval2 39637 aks4d1p1p4 40557 climrec 43918 |
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