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Theorem op2ndb 4804
 Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4209 to extract the first member and op2nda 4805 for an alternate version.) (Contributed by NM, 25-Nov-2003.)
Hypotheses
Ref Expression
cnvsn.1 𝐴 ∈ V
cnvsn.2 𝐵 ∈ V
Assertion
Ref Expression
op2ndb {⟨𝐴, 𝐵⟩} = 𝐵

Proof of Theorem op2ndb
StepHypRef Expression
1 cnvsn.1 . . . . . . 7 𝐴 ∈ V
2 cnvsn.2 . . . . . . 7 𝐵 ∈ V
31, 2cnvsn 4803 . . . . . 6 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
43inteqi 3619 . . . . 5 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
5 opexgOLD 3965 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ ∈ V)
62, 1, 5mp2an 402 . . . . . 6 𝐵, 𝐴⟩ ∈ V
76intsn 3650 . . . . 5 {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴
84, 7eqtri 2060 . . . 4 {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴
98inteqi 3619 . . 3 {⟨𝐴, 𝐵⟩} = 𝐵, 𝐴
109inteqi 3619 . 2 {⟨𝐴, 𝐵⟩} = 𝐵, 𝐴
112, 1op1stb 4209 . 2 𝐵, 𝐴⟩ = 𝐵
1210, 11eqtri 2060 1 {⟨𝐴, 𝐵⟩} = 𝐵
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {csn 3375  ⟨cop 3378  ∩ cint 3615  ◡ccnv 4344 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-int 3616  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353 This theorem is referenced by:  2ndval2  5783
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