Theorem 0we1 7586

Description: The empty set is a well-ordering of ordinal one. (Contributed by Mario Carneiro, 9-Feb-2015.)

Assertion

Ref	Expression
0we1	|- (/) We 1o

Proof of Theorem 0we1

Step	Hyp	Ref	Expression
1		br0 4701	. . 3 |- -. (/) (/) (/)
2		rel0 5243	. . . 4 |- Rel (/)
3		wesn 5190	. . . 4 |- ( Rel (/) -> ( (/) We { (/) } <-> -. (/) (/) (/) ) )
4	2, 3	ax-mp 5	. . 3 |- ( (/) We { (/) } <-> -. (/) (/) (/) )
5	1, 4	mpbir 221	. 2 |- (/) We { (/) }
6		df1o2 7572	. . 3 |- 1o = { (/) }
7		weeq2 5103	. . 3 |- ( 1o = { (/) } -> ( (/) We 1o <-> (/) We { (/) } ) )
8	6, 7	ax-mp 5	. 2 |- ( (/) We 1o <-> (/) We { (/) } )
9	5, 8	mpbir 221	1 |- (/) We 1o

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   (/)c0 3915   {csn 4177   class class class wbr 4653   We wwe 5072   Rel wrel 5119   1oc1o 7553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-suc 5729  df-1o 7560

This theorem is referenced by:  psr1tos  19559