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Theorem onntri13 7156
 Description: Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.)
Assertion
Ref Expression
onntri13 (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))

Proof of Theorem onntri13
StepHypRef Expression
1 nnral 2447 . 2 (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥 ∈ On ¬ ¬ ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
2 nnral 2447 . . 3 (¬ ¬ ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
32ralimi 2520 . 2 (∀𝑥 ∈ On ¬ ¬ ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
41, 3syl 14 1 (¬ ¬ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥 ∈ On ∀𝑦 ∈ On ¬ ¬ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∨ w3o 962  ∀wral 2435  Oncon0 4322 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-ral 2440  df-rex 2441 This theorem is referenced by:  onntri3or  7163
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