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Theorem sneqr 3559
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 3430 . . 3 𝐴 ∈ {𝐴}
3 eleq2 2117 . . 3 ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵}))
42, 3mpbii 140 . 2 ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵})
51elsn 3419 . 2 (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)
64, 5sylib 131 1 ({𝐴} = {𝐵} → 𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  Vcvv 2574  {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sn 3409
This theorem is referenced by:  sneqrg  3561  opth1  4001
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