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Theorem opeqsn 4270
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 3792 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43eqeq1i 2197 . 2 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶})
51snex 4203 . . 3 {𝐴} ∈ V
6 prexg 4229 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
71, 2, 6mp2an 426 . . 3 {𝐴, 𝐵} ∈ V
8 opeqsn.3 . . 3 𝐶 ∈ V
95, 7, 8preqsn 3790 . 2 ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶))
10 eqcom 2191 . . . . 5 ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
111, 2, 1preqsn 3790 . . . . 5 ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐵𝐵 = 𝐴))
12 eqcom 2191 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
1312anbi2i 457 . . . . . 6 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ (𝐴 = 𝐵𝐴 = 𝐵))
14 anidm 396 . . . . . 6 ((𝐴 = 𝐵𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
1513, 14bitri 184 . . . . 5 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
1610, 11, 153bitri 206 . . . 4 ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵)
1716anbi1i 458 . . 3 (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶))
18 dfsn2 3621 . . . . . . 7 {𝐴} = {𝐴, 𝐴}
19 preq2 3685 . . . . . . 7 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
2018, 19eqtr2id 2235 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
2120eqeq1d 2198 . . . . 5 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
22 eqcom 2191 . . . . 5 ({𝐴} = 𝐶𝐶 = {𝐴})
2321, 22bitrdi 196 . . . 4 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶𝐶 = {𝐴}))
2423pm5.32i 454 . . 3 ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
2517, 24bitri 184 . 2 (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
264, 9, 253bitri 206 1 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1364  wcel 2160  Vcvv 2752  {csn 3607  {cpr 3608  cop 3610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616
This theorem is referenced by:  relop  4795
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