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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
StepHypRef Expression
1 br0 4631 . . 3 ¬ ∅∅∅
2 rel0 5166 . . . 4 Rel ∅
3 wesn 5113 . . . 4 (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅))
42, 3ax-mp 5 . . 3 (∅ We {∅} ↔ ¬ ∅∅∅)
51, 4mpbir 220 . 2 ∅ We {∅}
6 df1o2 7459 . . 3 1𝑜 = {∅}
7 weeq2 5027 . . 3 (1𝑜 = {∅} → (∅ We 1𝑜 ↔ ∅ We {∅}))
86, 7ax-mp 5 . 2 (∅ We 1𝑜 ↔ ∅ We {∅})
95, 8mpbir 220 1 ∅ We 1𝑜
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475  ∅c0 3874  {csn 4125   class class class wbr 4583   We wwe 4996  Rel wrel 5043  1𝑜c1o 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
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