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Theorem snopeqop 4894
 Description: Equivalence for an ordered pair equal to a singleton of an ordered pair. (Contributed by AV, 18-Sep-2020.)
Hypotheses
Ref Expression
snopeqop.a 𝐴 ∈ V
snopeqop.b 𝐵 ∈ V
snopeqop.c 𝐶 ∈ V
snopeqop.d 𝐷 ∈ V
Assertion
Ref Expression
snopeqop ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}))

Proof of Theorem snopeqop
StepHypRef Expression
1 snopeqop.a . . . 4 𝐴 ∈ V
2 snopeqop.b . . . 4 𝐵 ∈ V
3 snopeqop.c . . . 4 𝐶 ∈ V
41, 2, 3opeqsn 4892 . . 3 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
54anbi2i 726 . 2 ((𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶}) ↔ (𝐶 = 𝐷 ∧ (𝐴 = 𝐵𝐶 = {𝐴})))
6 eqcom 2617 . . 3 ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = {⟨𝐴, 𝐵⟩})
7 snopeqop.d . . . 4 𝐷 ∈ V
8 opex 4859 . . . 4 𝐴, 𝐵⟩ ∈ V
93, 7, 8opeqsn 4892 . . 3 (⟨𝐶, 𝐷⟩ = {⟨𝐴, 𝐵⟩} ↔ (𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶}))
106, 9bitri 263 . 2 ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶}))
11 3anan12 1044 . 2 ((𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}) ↔ (𝐶 = 𝐷 ∧ (𝐴 = 𝐵𝐶 = {𝐴})))
125, 10, 113bitr4i 291 1 ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵𝐶 = 𝐷𝐶 = {𝐴}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132 This theorem is referenced by:  funopsn  6319  funsneqopsn  6322
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