Theorem 0we1 8443

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 5149	. . 3 ⊢ ¬ ∅∅∅
2		rel0 5756	. . . 4 ⊢ Rel ∅
3		wesn 5721	. . . 4 ⊢ (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅))
4	2, 3	ax-mp 5	. . 3 ⊢ (∅ We {∅} ↔ ¬ ∅∅∅)
5	1, 4	mpbir 231	. 2 ⊢ ∅ We {∅}
6		df1o2 8414	. . 3 ⊢ 1o = {∅}
7		weeq2 5620	. . 3 ⊢ (1o = {∅} → (∅ We 1o ↔ ∅ We {∅}))
8	6, 7	ax-mp 5	. 2 ⊢ (∅ We 1o ↔ ∅ We {∅})
9	5, 8	mpbir 231	1 ⊢ ∅ We 1o

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1542  ∅c0 4287  {csn 4582   class class class wbr 5100   We wwe 5584  Rel wrel 5637  1_oc1o 8400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-suc 6331  df-1o 8407

This theorem is referenced by:  psr1tos  22141