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Theorem opeqsn 4181
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 3711 . . 3 𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}
43eqeq1i 2148 . 2 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ {{𝐴}, {𝐴, 𝐵}} = {𝐶})
51snex 4116 . . 3 {𝐴} ∈ V
6 prexg 4140 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
71, 2, 6mp2an 423 . . 3 {𝐴, 𝐵} ∈ V
8 opeqsn.3 . . 3 𝐶 ∈ V
95, 7, 8preqsn 3709 . 2 ({{𝐴}, {𝐴, 𝐵}} = {𝐶} ↔ ({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶))
10 eqcom 2142 . . . . 5 ({𝐴} = {𝐴, 𝐵} ↔ {𝐴, 𝐵} = {𝐴})
111, 2, 1preqsn 3709 . . . . 5 ({𝐴, 𝐵} = {𝐴} ↔ (𝐴 = 𝐵𝐵 = 𝐴))
12 eqcom 2142 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
1312anbi2i 453 . . . . . 6 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ (𝐴 = 𝐵𝐴 = 𝐵))
14 anidm 394 . . . . . 6 ((𝐴 = 𝐵𝐴 = 𝐵) ↔ 𝐴 = 𝐵)
1513, 14bitri 183 . . . . 5 ((𝐴 = 𝐵𝐵 = 𝐴) ↔ 𝐴 = 𝐵)
1610, 11, 153bitri 205 . . . 4 ({𝐴} = {𝐴, 𝐵} ↔ 𝐴 = 𝐵)
1716anbi1i 454 . . 3 (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶))
18 dfsn2 3545 . . . . . . 7 {𝐴} = {𝐴, 𝐴}
19 preq2 3608 . . . . . . 7 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
2018, 19syl5req 2186 . . . . . 6 (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴})
2120eqeq1d 2149 . . . . 5 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶 ↔ {𝐴} = 𝐶))
22 eqcom 2142 . . . . 5 ({𝐴} = 𝐶𝐶 = {𝐴})
2321, 22syl6bb 195 . . . 4 (𝐴 = 𝐵 → ({𝐴, 𝐵} = 𝐶𝐶 = {𝐴}))
2423pm5.32i 450 . . 3 ((𝐴 = 𝐵 ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
2517, 24bitri 183 . 2 (({𝐴} = {𝐴, 𝐵} ∧ {𝐴, 𝐵} = 𝐶) ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
264, 9, 253bitri 205 1 (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵𝐶 = {𝐴}))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1332  wcel 1481  Vcvv 2689  {csn 3531  {cpr 3532  cop 3534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540
This theorem is referenced by:  relop  4696
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