Theorem we0 5635

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 5618	. 2 ⊢ 𝑅 Fr ∅
2		so0 5586	. 2 ⊢ 𝑅 Or ∅
3		df-we 5595	. 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
4	1, 2, 3	mpbir2an 719	1 ⊢ 𝑅 We ∅

Syntax hints:  ∅c0 4280   Or wor 5547   Fr wfr 5590   We wwe 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-po 5548  df-so 5549  df-fr 5593  df-we 5595

This theorem is referenced by:  ord0  6389  cantnf0  9620  cantnf  9638  wemapwe  9642  ltweuz  13964