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Theorem op2ndb 4880
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4273 to extract the first member and op2nda 4881 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 4879 . . . . . 6 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
43inteqi 3675 . . . . 5 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
52, 1opex 4030 . . . . . 6 𝐵, 𝐴⟩ ∈ V
65intsn 3706 . . . . 5 {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴
74, 6eqtri 2105 . . . 4 {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴
87inteqi 3675 . . 3 {⟨𝐴, 𝐵⟩} = 𝐵, 𝐴
98inteqi 3675 . 2 {⟨𝐴, 𝐵⟩} = 𝐵, 𝐴
102, 1op1stb 4273 . 2 𝐵, 𝐴⟩ = 𝐵
119, 10eqtri 2105 1 {⟨𝐴, 𝐵⟩} = 𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1287  wcel 1436  Vcvv 2615  {csn 3431  cop 3434   cint 3671  ccnv 4410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-int 3672  df-br 3821  df-opab 3875  df-xp 4417  df-rel 4418  df-cnv 4419
This theorem is referenced by:  2ndval2  5884
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