Theorem we0 2196

Description: Any relation is a well-ordering of the empty set.

Assertion

Ref	Expression
we0	⊢ R We ∅

Proof of Theorem we0

Step	Hyp	Ref	Expression
1		fr0 2179	. . 3 ⊢ R Fr ∅
2		so0 2153	. . 3 ⊢ R Or ∅
3	1, 2	pm3.2i 234	. 2 ⊢ (R Fr ∅ ∧ R Or ∅)
4		df-we 2186	. 2 ⊢ (R We ∅ ↔ (R Fr ∅ ∧ R Or ∅))
5	3, 4	mpbir 165	1 ⊢ R We ∅

Syntax hints:   ∧ wa 196  ∅c0 1707   Or wor 2059   Fr wfr 2061   We wwe 2062

This theorem is referenced by:  ord0 2276

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-po 2128  df-so 2138  df-fr 2169  df-we 2186

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