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Theorem we0 5079
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 5063 . 2 𝑅 Fr ∅
2 so0 5038 . 2 𝑅 Or ∅
3 df-we 5045 . 2 (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
41, 2, 3mpbir2an 954 1 𝑅 We ∅
Colors of variables: wff setvar class
Syntax hints:  c0 3897   Or wor 5004   Fr wfr 5040   We wwe 5042
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
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-in 3567  df-ss 3574  df-nul 3898  df-po 5005  df-so 5006  df-fr 5043  df-we 5045
This theorem is referenced by:  ord0  5746  cantnf0  8532  cantnf  8550  wemapwe  8554  ltweuz  12716
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