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Theorem we0 5338
Description: Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
Assertion
Ref Expression
we0 𝑅 We ∅

Proof of Theorem we0
StepHypRef Expression
1 fr0 5322 . 2 𝑅 Fr ∅
2 so0 5297 . 2 𝑅 Or ∅
3 df-we 5304 . 2 (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
41, 2, 3mpbir2an 704 1 𝑅 We ∅
Colors of variables: wff setvar class
Syntax hints:  c0 4145   Or wor 5263   Fr wfr 5299   We wwe 5301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-in 3806  df-ss 3813  df-nul 4146  df-po 5264  df-so 5265  df-fr 5302  df-we 5304
This theorem is referenced by:  ord0  6016  cantnf0  8850  cantnf  8868  wemapwe  8872  ltweuz  13056
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