Theorem we0 5552

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 5536	. 2 ⊢ 𝑅 Fr ∅
2		so0 5511	. 2 ⊢ 𝑅 Or ∅
3		df-we 5518	. 2 ⊢ (𝑅 We ∅ ↔ (𝑅 Fr ∅ ∧ 𝑅 Or ∅))
4	1, 2, 3	mpbir2an 709	1 ⊢ 𝑅 We ∅

Syntax hints:  ∅c0 4293   Or wor 5475   Fr wfr 5513   We wwe 5515

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-dif 3941  df-in 3945  df-ss 3954  df-nul 4294  df-po 5476  df-so 5477  df-fr 5516  df-we 5518

This theorem is referenced by:  ord0  6245  cantnf0  9140  cantnf  9158  wemapwe  9162  ltweuz  13332