Theorem we0 5654

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 5637	. 2 ⊢ 𝑅 Fr ∅
2		so0 5604	. 2 ⊢ 𝑅 Or ∅
3		df-we 5613	. 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
4	1, 2, 3	mpbir2an 711	1 ⊢ 𝑅 We ∅

Syntax hints:  ∅c0 4313   Or wor 5565   Fr wfr 5608   We wwe 5610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-po 5566  df-so 5567  df-fr 5611  df-we 5613

This theorem is referenced by:  ord0  6411  cantnf0  9694  cantnf  9712  wemapwe  9716  ltweuz  13984