Theorem 0we1 7473

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 4631	. . 3 ⊢ ¬ ∅∅∅
2		rel0 5166	. . . 4 ⊢ Rel ∅
3		wesn 5113	. . . 4 ⊢ (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅))
4	2, 3	ax-mp 5	. . 3 ⊢ (∅ We {∅} ↔ ¬ ∅∅∅)
5	1, 4	mpbir 220	. 2 ⊢ ∅ We {∅}
6		df1o2 7459	. . 3 ⊢ 1𝑜 = {∅}
7		weeq2 5027	. . 3 ⊢ (1𝑜 = {∅} → (∅ We 1𝑜 ↔ ∅ We {∅}))
8	6, 7	ax-mp 5	. 2 ⊢ (∅ We 1𝑜 ↔ ∅ We {∅})
9	5, 8	mpbir 220	1 ⊢ ∅ We 1𝑜

Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475  ∅c0 3874  {csn 4125   class class class wbr 4583   We wwe 4996  Rel wrel 5043  1_𝑜c1o 7440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-suc 5646  df-1o 7447

This theorem is referenced by:  psr1tos  19380