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Theorem opeqsng 5392
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 4800 . . 3 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
21eqeq1d 2823 . 2 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶}))
3 snex 5331 . . . 4 {𝐴} ∈ V
4 prex 5332 . . . 4 {𝐴, 𝐵} ∈ V
53, 4preqsn 4791 . . 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 2828 . . . . 5 ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
8 elex 3512 . . . . . . . 8 (𝐴𝑉𝐴 ∈ V)
98adantr 483 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → 𝐴 ∈ V)
10 elex 3512 . . . . . . . 8 (𝐵𝑊𝐵 ∈ V)
1110adantl 484 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → 𝐵 ∈ V)
129, 11preqsnd 4788 . . . . . 6 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐴𝐵 = 𝐴)))
13 simpr 487 . . . . . . . 8 ((𝐴 = 𝐴𝐵 = 𝐴) → 𝐵 = 𝐴)
14 eqid 2821 . . . . . . . . 9 𝐴 = 𝐴
1514jctl 526 . . . . . . . 8 (𝐵 = 𝐴 → (𝐴 = 𝐴𝐵 = 𝐴))
1613, 15impbii 211 . . . . . . 7 ((𝐴 = 𝐴𝐵 = 𝐴) ↔ 𝐵 = 𝐴)
17 eqcom 2828 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
1816, 17bitri 277 . . . . . 6 ((𝐴 = 𝐴𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
1912, 18syl6bb 289 . . . . 5 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ 𝐴 = 𝐵))
207, 19syl5bb 285 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵))
2120anbi1d 631 . . 3 ((𝐴𝑉𝐵𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶)))
22 dfsn2 4579 . . . . . . . 8 {𝐴} = {𝐴, 𝐴}
23 preq2 4669 . . . . . . . 8 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
2422, 23syl5req 2869 . . . . . . 7 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
2524eqeq1d 2823 . . . . . 6 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
26 eqcom 2828 . . . . . 6 ({𝐴} = 𝐶𝐶 = {𝐴})
2725, 26syl6bb 289 . . . . 5 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶𝐶 = {𝐴}))
2827a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶𝐶 = {𝐴})))
2928pm5.32d 579 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴})))
3021, 29bitrd 281 . 2 ((𝐴𝑉𝐵𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴})))
312, 6, 303bitrd 307 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  Vcvv 3494  {csn 4566  {cpr 4568  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:  opeqsn  5393  snopeqop  5395
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