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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 4839 | . 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 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sn 4627 |
| This theorem is referenced by: snsssn 4841 mosneq 4842 opth1 5480 propeqop 5512 opthwiener 5519 funsndifnop 7171 canth2 9170 axcc2lem 10476 hashge3el3dif 14526 dis2ndc 23468 axlowdim1 28974 bj-snsetex 36964 poimirlem13 37640 poimirlem14 37641 wopprc 43042 snen1g 43537 mnuprdlem2 44292 hoidmv1le 46609 fsetsnf1 47064 |
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