Theorem 0we1 8487

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 5160	. . 3 ⊢ ¬ ∅∅∅
2		rel0 5785	. . . 4 ⊢ Rel ∅
3		wesn 5750	. . . 4 ⊢ (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅))
4	2, 3	ax-mp 5	. . 3 ⊢ (∅ We {∅} ↔ ¬ ∅∅∅)
5	1, 4	mpbir 234	. 2 ⊢ ∅ We {∅}
6		df1o2 8456	. . 3 ⊢ 1o = {∅}
7		weeq2 5649	. . 3 ⊢ (1o = {∅} → (∅ We 1o ↔ ∅ We {∅}))
8	6, 7	ax-mp 5	. 2 ⊢ (∅ We 1o ↔ ∅ We {∅})
9	5, 8	mpbir 234	1 ⊢ ∅ We 1o

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1570  ∅c0 4286  {csn 4589   class class class wbr 5109   We wwe 5613  Rel wrel 5666  1_oc1o 8442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-suc 6366  df-1o 8449

This theorem is referenced by:  psr1tos  22349