Theorem 0we1 8497

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 5162	. . 3 ⊢ ¬ ∅∅∅
2		rel0 5787	. . . 4 ⊢ Rel ∅
3		wesn 5752	. . . 4 ⊢ (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅))
4	2, 3	ax-mp 5	. . 3 ⊢ (∅ We {∅} ↔ ¬ ∅∅∅)
5	1, 4	mpbir 234	. 2 ⊢ ∅ We {∅}
6		df1o2 8466	. . 3 ⊢ 1o = {∅}
7		weeq2 5651	. . 3 ⊢ (1o = {∅} → (∅ We 1o ↔ ∅ We {∅}))
8	6, 7	ax-mp 5	. 2 ⊢ (∅ We 1o ↔ ∅ We {∅})
9	5, 8	mpbir 234	1 ⊢ ∅ We 1o

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570  ∅c0 4286  {csn 4591   class class class wbr 5111   We wwe 5615  Rel wrel 5668  1_oc1o 8452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-suc 6370  df-1o 8459

This theorem is used by:  psr1tos  22401