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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 4370 | . 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 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-sn 4178 |
This theorem is referenced by: snsssn 4372 sneqrgOLD 4373 opth1 4944 propeqop 4970 opthwiener 4976 funsndifnop 6416 canth2 8113 axcc2lem 9258 hashge3el3dif 13268 dis2ndc 21263 axlowdim1 25839 bj-snsetex 32951 poimirlem13 33422 poimirlem14 33423 wopprc 37597 hoidmv1le 40808 |
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