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Theorem opeqsn 4927
Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
Hypotheses
Ref Expression
opeqsn.1 𝐴 ∈ V
opeqsn.2 𝐵 ∈ V
opeqsn.3 𝐶 ∈ V
Assertion
Ref Expression
opeqsn (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))

Proof of Theorem opeqsn
StepHypRef Expression
1 opeqsn.1 . . . 4 𝐴 ∈ V
2 opeqsn.2 . . . 4 𝐵 ∈ V
31, 2dfop 4369 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43eqeq1i 2626 . 2 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶})
5 snex 4869 . . 3 {𝐴} ∈ V
6 prex 4870 . . 3 {𝐴, 𝐵} ∈ V
7 opeqsn.3 . . 3 𝐶 ∈ V
85, 6, 7preqsn 4361 . 2 ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶))
9 eqcom 2628 . . . . 5 ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
101, 2, 1preqsn 4361 . . . . 5 ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐵𝐵 = 𝐴))
11 eqcom 2628 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
1211anbi2i 729 . . . . . 6 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ (𝐴 = 𝐵𝐴 = 𝐵))
13 anidm 675 . . . . . 6 ((𝐴 = 𝐵𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
1412, 13bitri 264 . . . . 5 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
159, 10, 143bitri 286 . . . 4 ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵)
1615anbi1i 730 . . 3 (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶))
17 dfsn2 4161 . . . . . . 7 {𝐴} = {𝐴, 𝐴}
18 preq2 4239 . . . . . . 7 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
1917, 18syl5req 2668 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
2019eqeq1d 2623 . . . . 5 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
21 eqcom 2628 . . . . 5 ({𝐴} = 𝐶𝐶 = {𝐴})
2220, 21syl6bb 276 . . . 4 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶𝐶 = {𝐴}))
2322pm5.32i 668 . . 3 ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
2416, 23bitri 264 . 2 (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
254, 8, 243bitri 286 1 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  {csn 4148  {cpr 4150  cop 4154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155
This theorem is referenced by:  snopeqop  4929  propeqop  4930  relop  5232
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