Theorem we0 5609

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 5592	. 2 ⊢ 𝑅 Fr ∅
2		so0 5560	. 2 ⊢ 𝑅 Or ∅
3		df-we 5569	. 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
4	1, 2, 3	mpbir2an 711	1 ⊢ 𝑅 We ∅

Syntax hints:  ∅c0 4281   Or wor 5521   Fr wfr 5564   We wwe 5566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-po 5522  df-so 5523  df-fr 5567  df-we 5569

This theorem is referenced by:  ord0  6356  cantnf0  9560  cantnf  9578  wemapwe  9582  ltweuz  13860