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Theorem op2ndb 5805
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5095 to extract the first member, op2nda 5806 for an alternate version, and op2nd 7375 for the preferred 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 5803 . . . . . 6 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
43inteqi 4637 . . . . 5 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
5 opex 5088 . . . . . 6 𝐵, 𝐴⟩ ∈ V
65intsn 4669 . . . . 5 {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴
74, 6eqtri 2787 . . . 4 {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴
87inteqi 4637 . . 3 {⟨𝐴, 𝐵⟩} = 𝐵, 𝐴
98inteqi 4637 . 2 {⟨𝐴, 𝐵⟩} = 𝐵, 𝐴
102, 1op1stb 5095 . 2 𝐵, 𝐴⟩ = 𝐵
119, 10eqtri 2787 1 {⟨𝐴, 𝐵⟩} = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  wcel 2155  Vcvv 3350  {csn 4334  cop 4340   cint 4633  ccnv 5276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-int 4634  df-br 4810  df-opab 4872  df-xp 5283  df-rel 5284  df-cnv 5285
This theorem is referenced by:  2ndval2  7384
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