Theorem 0we1 8432

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 5122	. . 3 ⊢ ¬ ∅∅∅
2		rel0 5743	. . . 4 ⊢ Rel ∅
3		wesn 5708	. . . 4 ⊢ (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅))
4	2, 3	ax-mp 5	. . 3 ⊢ (∅ We {∅} ↔ ¬ ∅∅∅)
5	1, 4	mpbir 232	. 2 ⊢ ∅ We {∅}
6		df1o2 8403	. . 3 ⊢ 1o = {∅}
7		weeq2 5607	. . 3 ⊢ (1o = {∅} → (∅ We 1o ↔ ∅ We {∅}))
8	6, 7	ax-mp 5	. 2 ⊢ (∅ We 1o ↔ ∅ We {∅})
9	5, 8	mpbir 232	1 ⊢ ∅ We 1o

Syntax hints:  ¬ wn 3   ↔ wb 207   = wceq 1547  ∅c0 4262  {csn 4556   class class class wbr 5073   We wwe 5571  Rel wrel 5624  1_oc1o 8389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-br 5074  df-opab 5136  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-suc 6317  df-1o 8396

This theorem is referenced by:  psr1tos  22175