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Theorem 0nelop 3976
Description: A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
0nelop ¬ ∅ A, B

Proof of Theorem 0nelop
StepHypRef Expression
1 id 19 . . . 4 (∅ A, B⟩ → ∅ A, B⟩)
2 oprcl 3564 . . . . 5 (∅ A, B⟩ → (A V B V))
3 dfopg 3538 . . . . 5 ((A V B V) → ⟨A, B⟩ = {{A}, {A, B}})
42, 3syl 14 . . . 4 (∅ A, B⟩ → ⟨A, B⟩ = {{A}, {A, B}})
51, 4eleqtrd 2113 . . 3 (∅ A, B⟩ → ∅ {{A}, {A, B}})
6 elpri 3387 . . 3 (∅ {{A}, {A, B}} → (∅ = {A} ∅ = {A, B}))
75, 6syl 14 . 2 (∅ A, B⟩ → (∅ = {A} ∅ = {A, B}))
82simpld 105 . . . . . 6 (∅ A, B⟩ → A V)
9 snnzg 3476 . . . . . 6 (A V → {A} ≠ ∅)
108, 9syl 14 . . . . 5 (∅ A, B⟩ → {A} ≠ ∅)
1110necomd 2285 . . . 4 (∅ A, B⟩ → ∅ ≠ {A})
12 prnzg 3483 . . . . . 6 (A V → {A, B} ≠ ∅)
138, 12syl 14 . . . . 5 (∅ A, B⟩ → {A, B} ≠ ∅)
1413necomd 2285 . . . 4 (∅ A, B⟩ → ∅ ≠ {A, B})
1511, 14jca 290 . . 3 (∅ A, B⟩ → (∅ ≠ {A} ∅ ≠ {A, B}))
16 neanior 2286 . . 3 ((∅ ≠ {A} ∅ ≠ {A, B}) ↔ ¬ (∅ = {A} ∅ = {A, B}))
1715, 16sylib 127 . 2 (∅ A, B⟩ → ¬ (∅ = {A} ∅ = {A, B}))
187, 17pm2.65i 567 1 ¬ ∅ A, B
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   wo 628   = wceq 1242   wcel 1390  wne 2201  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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-un 2916  df-nul 3219  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by:  0nelelxp  4316
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