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Theorem opprc 3561
 Description: Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opprc (¬ (A V B V) → ⟨A, B⟩ = ∅)

Proof of Theorem opprc
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 df-op 3376 . 2 A, B⟩ = {x ∣ (A V B V x {{A}, {A, B}})}
2 3simpa 900 . . . . 5 ((A V B V x {{A}, {A, B}}) → (A V B V))
32con3i 561 . . . 4 (¬ (A V B V) → ¬ (A V B V x {{A}, {A, B}}))
43alrimiv 1751 . . 3 (¬ (A V B V) → x ¬ (A V B V x {{A}, {A, B}}))
5 abeq0 3242 . . 3 ({x ∣ (A V B V x {{A}, {A, B}})} = ∅ ↔ x ¬ (A V B V x {{A}, {A, B}}))
64, 5sylibr 137 . 2 (¬ (A V B V) → {x ∣ (A V B V x {{A}, {A, B}})} = ∅)
71, 6syl5eq 2081 1 (¬ (A V B V) → ⟨A, B⟩ = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∧ w3a 884  ∀wal 1240   = wceq 1242   ∈ wcel 1390  {cab 2023  Vcvv 2551  ∅c0 3218  {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-in1 544  ax-in2 545  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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-nul 3219  df-op 3376 This theorem is referenced by:  opprc1  3562  opprc2  3563  ovprc  5482
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