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Mirrors > Home > MPE Home > Th. List > sneqr | Structured version Visualization version GIF version |
Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.) |
Ref | Expression |
---|---|
sneqr.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
sneqr | ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneqr.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sneqrg 4838 | . 2 ⊢ (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3462 {csn 4623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-sn 4624 |
This theorem is referenced by: snsssn 4840 mosneq 4841 opth1 5473 propeqop 5505 opthwiener 5512 funsndifnop 7157 canth2 9160 axcc2lem 10470 hashge3el3dif 14500 dis2ndc 23452 axlowdim1 28890 bj-snsetex 36683 poimirlem13 37347 poimirlem14 37348 wopprc 42725 snen1g 43228 mnuprdlem2 43984 hoidmv1le 46251 fsetsnf1 46703 |
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