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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 4770 | . 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 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-sn 4568 |
This theorem is referenced by: snsssn 4772 mosneq 4773 opth1 5367 propeqop 5397 opthwiener 5404 funsndifnop 6913 canth2 8670 axcc2lem 9858 hashge3el3dif 13845 dis2ndc 22068 axlowdim1 26745 bj-snsetex 34278 poimirlem13 34920 poimirlem14 34921 wopprc 39647 snen1g 39910 mnuprdlem2 40629 hoidmv1le 42896 |
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