Theorem we0 5620

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 5603	. 2 ⊢ 𝑅 Fr ∅
2		so0 5571	. 2 ⊢ 𝑅 Or ∅
3		df-we 5580	. 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
4	1, 2, 3	mpbir2an 712	1 ⊢ 𝑅 We ∅

Syntax hints:  ∅c0 4286   Or wor 5532   Fr wfr 5575   We wwe 5577

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-po 5533  df-so 5534  df-fr 5578  df-we 5580

This theorem is referenced by:  ord0  6372  cantnf0  9588  cantnf  9606  wemapwe  9610  ltweuz  13888