MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  op2ndb Structured version   Visualization version   GIF version

Theorem op2ndb 6247
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5476 to extract the first member, op2nda 6248 for an alternate version, and op2nd 8023 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 6246 . . . . . 6 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
43inteqi 4950 . . . . 5 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
5 opex 5469 . . . . . 6 𝐵, 𝐴⟩ ∈ V
65intsn 4984 . . . . 5 {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴
74, 6eqtri 2765 . . . 4 {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴
87inteqi 4950 . . 3 {⟨𝐴, 𝐵⟩} = 𝐵, 𝐴
98inteqi 4950 . 2 {⟨𝐴, 𝐵⟩} = 𝐵, 𝐴
102, 1op1stb 5476 . 2 𝐵, 𝐴⟩ = 𝐵
119, 10eqtri 2765 1 {⟨𝐴, 𝐵⟩} = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632   cint 4946  ccnv 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-int 4947  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693
This theorem is referenced by:  2ndval2  8032
  Copyright terms: Public domain W3C validator