Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnvsng Structured version   GIF version

Theorem cnvsng 4749
 Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
Assertion
Ref Expression
cnvsng ((A 𝑉 B 𝑊) → {⟨A, B⟩} = {⟨B, A⟩})

Proof of Theorem cnvsng
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3540 . . . . 5 (x = A → ⟨x, y⟩ = ⟨A, y⟩)
21sneqd 3380 . . . 4 (x = A → {⟨x, y⟩} = {⟨A, y⟩})
32cnveqd 4454 . . 3 (x = A{⟨x, y⟩} = {⟨A, y⟩})
4 opeq2 3541 . . . 4 (x = A → ⟨y, x⟩ = ⟨y, A⟩)
54sneqd 3380 . . 3 (x = A → {⟨y, x⟩} = {⟨y, A⟩})
63, 5eqeq12d 2051 . 2 (x = A → ({⟨x, y⟩} = {⟨y, x⟩} ↔ {⟨A, y⟩} = {⟨y, A⟩}))
7 opeq2 3541 . . . . 5 (y = B → ⟨A, y⟩ = ⟨A, B⟩)
87sneqd 3380 . . . 4 (y = B → {⟨A, y⟩} = {⟨A, B⟩})
98cnveqd 4454 . . 3 (y = B{⟨A, y⟩} = {⟨A, B⟩})
10 opeq1 3540 . . . 4 (y = B → ⟨y, A⟩ = ⟨B, A⟩)
1110sneqd 3380 . . 3 (y = B → {⟨y, A⟩} = {⟨B, A⟩})
129, 11eqeq12d 2051 . 2 (y = B → ({⟨A, y⟩} = {⟨y, A⟩} ↔ {⟨A, B⟩} = {⟨B, A⟩}))
13 vex 2554 . . 3 x V
14 vex 2554 . . 3 y V
1513, 14cnvsn 4746 . 2 {⟨x, y⟩} = {⟨y, x⟩}
166, 12, 15vtocl2g 2611 1 ((A 𝑉 B 𝑊) → {⟨A, B⟩} = {⟨B, A⟩})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  {csn 3367  ⟨cop 3370  ◡ccnv 4287 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296 This theorem is referenced by:  opswapg  4750  funsng  4889
 Copyright terms: Public domain W3C validator