Theorem we0 5514

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 5498	. 2 ⊢ 𝑅 Fr ∅
2		so0 5473	. 2 ⊢ 𝑅 Or ∅
3		df-we 5480	. 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
4	1, 2, 3	mpbir2an 710	1 ⊢ 𝑅 We ∅

Syntax hints:  ∅c0 4243   Or wor 5437   Fr wfr 5475   We wwe 5477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-nul 4244  df-po 5438  df-so 5439  df-fr 5478  df-we 5480

This theorem is referenced by:  ord0  6211  cantnf0  9122  cantnf  9140  wemapwe  9144  ltweuz  13324