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Theorem opeqsn 5393
 Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.)
Hypotheses
Ref Expression
opeqsn.1 𝐴 ∈ V
opeqsn.2 𝐵 ∈ V
Assertion
Ref Expression
opeqsn (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))

Proof of Theorem opeqsn
StepHypRef Expression
1 opeqsn.1 . 2 𝐴 ∈ V
2 opeqsn.2 . 2 𝐵 ∈ V
3 opeqsng 5392 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴})))
41, 2, 3mp2an 690 1 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494  {csn 4566  ⟨cop 4572 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573 This theorem is referenced by:  snopeqop  5395  propeqop  5396  relop  5720
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