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Theorem 0nelxpOLD 5104
Description: Obsolete proof of 0nelxp 5103 as of 13-Aug-2021. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
0nelxpOLD ¬ ∅ ∈ (𝐴 × 𝐵)

Proof of Theorem 0nelxpOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3189 . . . . . 6 𝑥 ∈ V
2 vex 3189 . . . . . 6 𝑦 ∈ V
31, 2opnzi 4903 . . . . 5 𝑥, 𝑦⟩ ≠ ∅
4 simpl 473 . . . . . . 7 ((∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ∅ = ⟨𝑥, 𝑦⟩)
54eqcomd 2627 . . . . . 6 ((∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → ⟨𝑥, 𝑦⟩ = ∅)
65necon3ai 2815 . . . . 5 (⟨𝑥, 𝑦⟩ ≠ ∅ → ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
73, 6ax-mp 5 . . . 4 ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
87nex 1728 . . 3 ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
98nex 1728 . 2 ¬ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵))
10 elxp 5091 . 2 (∅ ∈ (𝐴 × 𝐵) ↔ ∃𝑥𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
119, 10mtbir 313 1 ¬ ∅ ∈ (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  c0 3891  cop 4154   × cxp 5072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-opab 4674  df-xp 5080
This theorem is referenced by: (None)
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