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Theorem 0nelxp 4315
 Description: The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
0nelxp ¬ ∅ (A × B)

Proof of Theorem 0nelxp
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . 6 x V
2 vex 2554 . . . . . 6 y V
31, 2opnzi 3963 . . . . 5 x, y⟩ ≠ ∅
4 simpl 102 . . . . . . 7 ((∅ = ⟨x, y (x A y B)) → ∅ = ⟨x, y⟩)
54eqcomd 2042 . . . . . 6 ((∅ = ⟨x, y (x A y B)) → ⟨x, y⟩ = ∅)
65necon3ai 2248 . . . . 5 (⟨x, y⟩ ≠ ∅ → ¬ (∅ = ⟨x, y (x A y B)))
73, 6ax-mp 7 . . . 4 ¬ (∅ = ⟨x, y (x A y B))
87nex 1386 . . 3 ¬ y(∅ = ⟨x, y (x A y B))
98nex 1386 . 2 ¬ xy(∅ = ⟨x, y (x A y B))
10 elxp 4305 . 2 (∅ (A × B) ↔ xy(∅ = ⟨x, y (x A y B)))
119, 10mtbir 595 1 ¬ ∅ (A × B)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390   ≠ wne 2201  ∅c0 3218  ⟨cop 3370   × cxp 4286 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-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-fal 1248  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-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294 This theorem is referenced by:  dmsn0  4731  nfunv  4876  reldmtpos  5809  dmtpos  5812  0ncn  6709
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