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Theorem opeqsn 5158
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 5157 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴})))
41, 2, 3mp2an 684 1 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  {csn 4368  cop 4374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375
This theorem is referenced by:  snopeqop  5161  propeqop  5163  relop  5476
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