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Theorem we0 5672
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 5656 . 2 𝑅 Fr ∅
2 so0 5625 . 2 𝑅 Or ∅
3 df-we 5634 . 2 (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
41, 2, 3mpbir2an 710 1 𝑅 We ∅
Colors of variables: wff setvar class
Syntax hints:  c0 4323   Or wor 5588   Fr wfr 5629   We wwe 5631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-po 5589  df-so 5590  df-fr 5632  df-we 5634
This theorem is referenced by:  ord0  6418  cantnf0  9670  cantnf  9688  wemapwe  9692  ltweuz  13926
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