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Theorem sneqr 3772
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.)
Hypothesis
Ref Expression
sneqr.1 𝐴 ∈ V
Assertion
Ref Expression
sneqr ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Proof of Theorem sneqr
StepHypRef Expression
1 sneqr.1 . . . 4 𝐴 ∈ V
21snid 3635 . . 3 𝐴 ∈ {𝐴}
3 eleq2 2251 . . 3 ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵}))
42, 3mpbii 148 . 2 ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵})
51elsn 3620 . 2 (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)
64, 5sylib 122 1 ({𝐴} = {𝐵} → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wcel 2158  Vcvv 2749  {csn 3604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sn 3610
This theorem is referenced by:  sneqrg  3774  opth1  4248  cc2lem  7279
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