Theorem 0we1 7933

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 4978	. . 3 ⊢ ¬ ∅∅∅
2		rel0 5522	. . . 4 ⊢ Rel ∅
3		wesn 5490	. . . 4 ⊢ (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅))
4	2, 3	ax-mp 5	. . 3 ⊢ (∅ We {∅} ↔ ¬ ∅∅∅)
5	1, 4	mpbir 223	. 2 ⊢ ∅ We {∅}
6		df1o2 7918	. . 3 ⊢ 1o = {∅}
7		weeq2 5396	. . 3 ⊢ (1o = {∅} → (∅ We 1o ↔ ∅ We {∅}))
8	6, 7	ax-mp 5	. 2 ⊢ (∅ We 1o ↔ ∅ We {∅})
9	5, 8	mpbir 223	1 ⊢ ∅ We 1o

Syntax hints:  ¬ wn 3   ↔ wb 198   = wceq 1507  ∅c0 4179  {csn 4441   class class class wbr 4929   We wwe 5365  Rel wrel 5412  1_oc1o 7898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-suc 6035  df-1o 7905

This theorem is referenced by:  psr1tos  20060