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Theorem we0 5564
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 5548 . 2 𝑅 Fr ∅
2 so0 5522 . 2 𝑅 Or ∅
3 df-we 5529 . 2 (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
41, 2, 3mpbir2an 711 1 𝑅 We ∅
Colors of variables: wff setvar class
Syntax hints:  c0 4254   Or wor 5485   Fr wfr 5524   We wwe 5526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-po 5486  df-so 5487  df-fr 5527  df-we 5529
This theorem is referenced by:  ord0  6286  cantnf0  9320  cantnf  9338  wemapwe  9342  ltweuz  13566
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