Theorem we0 5338

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 5322	. 2 ⊢ 𝑅 Fr ∅
2		so0 5297	. 2 ⊢ 𝑅 Or ∅
3		df-we 5304	. 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
4	1, 2, 3	mpbir2an 704	1 ⊢ 𝑅 We ∅

Syntax hints:  ∅c0 4145   Or wor 5263   Fr wfr 5299   We wwe 5301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-in 3806  df-ss 3813  df-nul 4146  df-po 5264  df-so 5265  df-fr 5302  df-we 5304

This theorem is referenced by:  ord0  6016  cantnf0  8850  cantnf  8868  wemapwe  8872  ltweuz  13056