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Theorem op2ndb 6179
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5428 to extract the first member, op2nda 6180 for an alternate version, and op2nd 7929 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 6178 . . . . . 6 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
43inteqi 4911 . . . . 5 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
5 opex 5421 . . . . . 6 𝐵, 𝐴⟩ ∈ V
65intsn 4947 . . . . 5 {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴
74, 6eqtri 2764 . . . 4 {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴
87inteqi 4911 . . 3 {⟨𝐴, 𝐵⟩} = 𝐵, 𝐴
98inteqi 4911 . 2 {⟨𝐴, 𝐵⟩} = 𝐵, 𝐴
102, 1op1stb 5428 . 2 𝐵, 𝐴⟩ = 𝐵
119, 10eqtri 2764 1 {⟨𝐴, 𝐵⟩} = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wcel 2106  Vcvv 3445  {csn 4586  cop 4592   cint 4907  ccnv 5632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-int 4908  df-br 5106  df-opab 5168  df-xp 5639  df-rel 5640  df-cnv 5641
This theorem is referenced by:  2ndval2  7938
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