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Theorem 0nelelxp 4316
Description: A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)
Assertion
Ref Expression
0nelelxp (𝐶 (A × B) → ¬ ∅ 𝐶)

Proof of Theorem 0nelelxp
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 4305 . 2 (𝐶 (A × B) ↔ xy(𝐶 = ⟨x, y (x A y B)))
2 0nelop 3976 . . . 4 ¬ ∅ x, y
3 simpl 102 . . . . 5 ((𝐶 = ⟨x, y (x A y B)) → 𝐶 = ⟨x, y⟩)
43eleq2d 2104 . . . 4 ((𝐶 = ⟨x, y (x A y B)) → (∅ 𝐶 ↔ ∅ x, y⟩))
52, 4mtbiri 599 . . 3 ((𝐶 = ⟨x, y (x A y B)) → ¬ ∅ 𝐶)
65exlimivv 1773 . 2 (xy(𝐶 = ⟨x, y (x A y B)) → ¬ ∅ 𝐶)
71, 6sylbi 114 1 (𝐶 (A × B) → ¬ ∅ 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  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-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:  dmsn0el  4733
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