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Theorem sneqr 3687
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 3556 . . 3 𝐴 ∈ {𝐴}
3 eleq2 2203 . . 3 ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵}))
42, 3mpbii 147 . 2 ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵})
51elsn 3543 . 2 (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)
64, 5sylib 121 1 ({𝐴} = {𝐵} → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  Vcvv 2686  {csn 3527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sn 3533
This theorem is referenced by:  sneqrg  3689  opth1  4158  cc2lem  7081
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