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Theorem preqsn 3574
 Description: Equivalence for a pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
Hypotheses
Ref Expression
preqsn.1 𝐴 ∈ V
preqsn.2 𝐵 ∈ V
preqsn.3 𝐶 ∈ V
Assertion
Ref Expression
preqsn ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))

Proof of Theorem preqsn
StepHypRef Expression
1 dfsn2 3417 . . 3 {𝐶} = {𝐶, 𝐶}
21eqeq2i 2066 . 2 ({𝐴, 𝐵} = {𝐶} ↔ {𝐴, 𝐵} = {𝐶, 𝐶})
3 preqsn.1 . . . 4 𝐴 ∈ V
4 preqsn.2 . . . 4 𝐵 ∈ V
5 preqsn.3 . . . 4 𝐶 ∈ V
63, 4, 5, 5preq12b 3569 . . 3 ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ ((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)))
7 oridm 684 . . . 4 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐶))
8 eqtr3 2075 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐴 = 𝐵)
9 simpr 107 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐶) → 𝐵 = 𝐶)
108, 9jca 294 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐶) → (𝐴 = 𝐵𝐵 = 𝐶))
11 eqtr 2073 . . . . . 6 ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 𝐶)
12 simpr 107 . . . . . 6 ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐵 = 𝐶)
1311, 12jca 294 . . . . 5 ((𝐴 = 𝐵𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐶))
1410, 13impbii 121 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐶) ↔ (𝐴 = 𝐵𝐵 = 𝐶))
157, 14bitri 177 . . 3 (((𝐴 = 𝐶𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐶)) ↔ (𝐴 = 𝐵𝐵 = 𝐶))
166, 15bitri 177 . 2 ({𝐴, 𝐵} = {𝐶, 𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))
172, 16bitri 177 1 ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐵𝐵 = 𝐶))
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   ↔ wb 102   ∨ wo 639   = wceq 1259   ∈ wcel 1409  Vcvv 2574  {csn 3403  {cpr 3404 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410 This theorem is referenced by:  opeqsn  4017  relop  4514
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