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

Theorem opm 3962
 Description: An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.)
Assertion
Ref Expression
opm (x x A, B⟩ ↔ (A V B V))
Distinct variable groups:   x,A   x,B

Proof of Theorem opm
StepHypRef Expression
1 df-op 3376 . . . . 5 A, B⟩ = {x ∣ (A V B V x {{A}, {A, B}})}
21eleq2i 2101 . . . 4 (x A, B⟩ ↔ x {x ∣ (A V B V x {{A}, {A, B}})})
32exbii 1493 . . 3 (x x A, B⟩ ↔ x x {x ∣ (A V B V x {{A}, {A, B}})})
4 abid 2025 . . . 4 (x {x ∣ (A V B V x {{A}, {A, B}})} ↔ (A V B V x {{A}, {A, B}}))
54exbii 1493 . . 3 (x x {x ∣ (A V B V x {{A}, {A, B}})} ↔ x(A V B V x {{A}, {A, B}}))
63, 5bitri 173 . 2 (x x A, B⟩ ↔ x(A V B V x {{A}, {A, B}}))
7 19.42v 1783 . . 3 (x((A V B V) x {{A}, {A, B}}) ↔ ((A V B V) x x {{A}, {A, B}}))
8 df-3an 886 . . . 4 ((A V B V x {{A}, {A, B}}) ↔ ((A V B V) x {{A}, {A, B}}))
98exbii 1493 . . 3 (x(A V B V x {{A}, {A, B}}) ↔ x((A V B V) x {{A}, {A, B}}))
10 df-3an 886 . . 3 ((A V B V x x {{A}, {A, B}}) ↔ ((A V B V) x x {{A}, {A, B}}))
117, 9, 103bitr4ri 202 . 2 ((A V B V x x {{A}, {A, B}}) ↔ x(A V B V x {{A}, {A, B}}))
12 3simpa 900 . . 3 ((A V B V x x {{A}, {A, B}}) → (A V B V))
13 id 19 . . . 4 ((A V B V) → (A V B V))
14 snexgOLD 3926 . . . . . 6 (A V → {A} V)
1514adantr 261 . . . . 5 ((A V B V) → {A} V)
16 prmg 3480 . . . . 5 ({A} V → x x {{A}, {A, B}})
1715, 16syl 14 . . . 4 ((A V B V) → x x {{A}, {A, B}})
1813, 17, 10sylanbrc 394 . . 3 ((A V B V) → (A V B V x x {{A}, {A, B}}))
1912, 18impbii 117 . 2 ((A V B V x x {{A}, {A, B}}) ↔ (A V B V))
206, 11, 193bitr2i 197 1 (x x A, B⟩ ↔ (A V B V))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∧ w3a 884  ∃wex 1378   ∈ wcel 1390  {cab 2023  Vcvv 2551  {csn 3367  {cpr 3368  ⟨cop 3370 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 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376 This theorem is referenced by:  opnzi  3963  opeqex  3977
 Copyright terms: Public domain W3C validator