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Theorem opeqsng 5157
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 4591 . . 3 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
21eqeq1d 2801 . 2 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶}))
3 snex 5099 . . . 4 {𝐴} ∈ V
4 prex 5100 . . . 4 {𝐴, 𝐵} ∈ V
53, 4preqsn 4581 . . 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 2806 . . . . 5 ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
8 elex 3400 . . . . . . . 8 (𝐴𝑉𝐴 ∈ V)
98adantr 473 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → 𝐴 ∈ V)
10 elex 3400 . . . . . . . 8 (𝐵𝑊𝐵 ∈ V)
1110adantl 474 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → 𝐵 ∈ V)
129, 11preqsnd 4577 . . . . . 6 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐴𝐵 = 𝐴)))
13 simpr 478 . . . . . . . 8 ((𝐴 = 𝐴𝐵 = 𝐴) → 𝐵 = 𝐴)
14 eqid 2799 . . . . . . . . 9 𝐴 = 𝐴
1514jctl 520 . . . . . . . 8 (𝐵 = 𝐴 → (𝐴 = 𝐴𝐵 = 𝐴))
1613, 15impbii 201 . . . . . . 7 ((𝐴 = 𝐴𝐵 = 𝐴) ↔ 𝐵 = 𝐴)
17 eqcom 2806 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
1816, 17bitri 267 . . . . . 6 ((𝐴 = 𝐴𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
1912, 18syl6bb 279 . . . . 5 ((𝐴𝑉𝐵𝑊) → ({𝐴, 𝐵} = {𝐴} ↔ 𝐴 = 𝐵))
207, 19syl5bb 275 . . . 4 ((𝐴𝑉𝐵𝑊) → ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵))
2120anbi1d 624 . . 3 ((𝐴𝑉𝐵𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶)))
22 dfsn2 4381 . . . . . . . 8 {𝐴} = {𝐴, 𝐴}
23 preq2 4458 . . . . . . . 8 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
2422, 23syl5req 2846 . . . . . . 7 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
2524eqeq1d 2801 . . . . . 6 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
26 eqcom 2806 . . . . . 6 ({𝐴} = 𝐶𝐶 = {𝐴})
2725, 26syl6bb 279 . . . . 5 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶𝐶 = {𝐴}))
2827a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶𝐶 = {𝐴})))
2928pm5.32d 573 . . 3 ((𝐴𝑉𝐵𝑊) → ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴})))
3021, 29bitrd 271 . 2 ((𝐴𝑉𝐵𝑊) → (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴})))
312, 6, 303bitrd 297 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  {csn 4368  {cpr 4370  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:  opeqsn  5158  snopeqop  5161
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