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Theorem opeqsng 5504
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.)
Assertion
Ref Expression
opeqsng ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴})))

Proof of Theorem opeqsng
StepHypRef Expression
1 dfopg 4872 . . 3 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
21eqeq1d 2735 . 2 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶}))
3 snex 5432 . . . 4 {𝐴} ∈ V
4 prex 5433 . . . 4 {𝐴, 𝐵} ∈ V
53, 4preqsn 4863 . . 3 ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶))
65a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶)))
7 eqcom 2740 . . . . 5 ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
8 simpl 484 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
9 simpr 486 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
108, 9preqsnd 4860 . . . . . 6 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐴𝐵 = 𝐴)))
11 simpr 486 . . . . . . . 8 ((𝐴 = 𝐴𝐵 = 𝐴) → 𝐵 = 𝐴)
12 eqid 2733 . . . . . . . . 9 𝐴 = 𝐴
1312jctl 525 . . . . . . . 8 (𝐵 = 𝐴 → (𝐴 = 𝐴𝐵 = 𝐴))
1411, 13impbii 208 . . . . . . 7 ((𝐴 = 𝐴𝐵 = 𝐴) ↔ 𝐵 = 𝐴)
15 eqcom 2740 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
1614, 15bitri 275 . . . . . 6 ((𝐴 = 𝐴𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
1710, 16bitrdi 287 . . . . 5 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ 𝐴 = 𝐵))
187, 17bitrid 283 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵))
1918anbi1d 631 . . 3 ((𝐴𝑉𝐵𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶)))
20 dfsn2 4642 . . . . . . . 8 {𝐴} = {𝐴, 𝐴}
21 preq2 4739 . . . . . . . 8 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
2220, 21eqtr2id 2786 . . . . . . 7 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
2322eqeq1d 2735 . . . . . 6 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
24 eqcom 2740 . . . . . 6 ({𝐴} = 𝐶𝐶 = {𝐴})
2523, 24bitrdi 287 . . . . 5 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶𝐶 = {𝐴}))
2625a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶𝐶 = {𝐴})))
2726pm5.32d 578 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴})))
2819, 27bitrd 279 . 2 ((𝐴𝑉𝐵𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴})))
292, 6, 283bitrd 305 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {csn 4629  {cpr 4631  cop 4635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636
This theorem is referenced by:  opeqsn  5505  snopeqop  5507
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