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 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}))

Proof of Theorem snopeqop

Step	Hyp	Ref	Expression
1		snopeqop.a	. . . 4 ⊢ 𝐴 ∈ V
2		snopeqop.b	. . . 4 ⊢ 𝐵 ∈ V
3		snopeqop.c	. . . 4 ⊢ 𝐶 ∈ V
4	1, 2, 3	opeqsn 4892	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))
5	4	anbi2i 726	. 2 ⊢ ((𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶}) ↔ (𝐶 = 𝐷 ∧ (𝐴 = 𝐵 ∧ 𝐶 = {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
9	3, 7, 8	opeqsn 4892	. . 3 ⊢ (⟨𝐶, 𝐷⟩ = {⟨𝐴, 𝐵⟩} ↔ (𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶}))
10	6, 9	bitri 263	. 2 ⊢ ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐶 = 𝐷 ∧ ⟨𝐴, 𝐵⟩ = {𝐶}))
11		3anan12 1044	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}) ↔ (𝐶 = 𝐷 ∧ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))
12	5, 10, 11	3bitr4i 291	1 ⊢ ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}))

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