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Theorem we0 5543
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 5527 . 2 𝑅 Fr ∅
2 so0 5502 . 2 𝑅 Or ∅
3 df-we 5509 . 2 (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
41, 2, 3mpbir2an 709 1 𝑅 We ∅
Colors of variables: wff setvar class
Syntax hints:  c0 4289   Or wor 5466   Fr wfr 5504   We wwe 5506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-dif 3937  df-in 3941  df-ss 3950  df-nul 4290  df-po 5467  df-so 5468  df-fr 5507  df-we 5509
This theorem is referenced by:  ord0  6236  cantnf0  9130  cantnf  9148  wemapwe  9152  ltweuz  13321
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