Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4764 | . 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 1528 ∈ wcel 2105 Vcvv 3495 {csn 4559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-sn 4560 |
This theorem is referenced by: snsssn 4766 mosneq 4767 opth1 5359 propeqop 5389 opthwiener 5396 funsndifnop 6906 canth2 8659 axcc2lem 9847 hashge3el3dif 13834 dis2ndc 21998 axlowdim1 26673 bj-snsetex 34173 poimirlem13 34787 poimirlem14 34788 wopprc 39507 snen1g 39770 mnuprdlem2 40489 hoidmv1le 42757 |
Copyright terms: Public domain | W3C validator |