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Theorem elpr2 3425
Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)
Hypotheses
Ref Expression
elpr2.1 𝐵 ∈ V
elpr2.2 𝐶 ∈ V
Assertion
Ref Expression
elpr2 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))

Proof of Theorem elpr2
StepHypRef Expression
1 elprg 3423 . . 3 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
21ibi 169 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
3 elpr2.1 . . . . . 6 𝐵 ∈ V
4 eleq1 2116 . . . . . 6 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
53, 4mpbiri 161 . . . . 5 (𝐴 = 𝐵𝐴 ∈ V)
6 elpr2.2 . . . . . 6 𝐶 ∈ V
7 eleq1 2116 . . . . . 6 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
86, 7mpbiri 161 . . . . 5 (𝐴 = 𝐶𝐴 ∈ V)
95, 8jaoi 646 . . . 4 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐴 ∈ V)
10 elprg 3423 . . . 4 (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
119, 10syl 14 . . 3 ((𝐴 = 𝐵𝐴 = 𝐶) → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶)))
1211ibir 170 . 2 ((𝐴 = 𝐵𝐴 = 𝐶) → 𝐴 ∈ {𝐵, 𝐶})
132, 12impbii 121 1 (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵𝐴 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wb 102  wo 639   = wceq 1259  wcel 1409  Vcvv 2574  {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:  elxr  8797
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