Theorem we0 5033

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

Step	Hyp	Ref	Expression
1		fr0 5017	. 2 ⊢ 𝑅 Fr ∅
2		so0 4992	. 2 ⊢ 𝑅 Or ∅
3		df-we 4999	. 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
4	1, 2, 3	mpbir2an 957	1 ⊢ 𝑅 We ∅

Syntax hints:  ∅c0 3874   Or wor 4958   Fr wfr 4994   We wwe 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-po 4959  df-so 4960  df-fr 4997  df-we 4999

This theorem is referenced by:  ord0  5694  cantnf0  8455  cantnf  8473  wemapwe  8477  ltweuz  12622