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Theorem 0we1 8117
 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 5080 . . 3 ¬ ∅∅∅
2 rel0 5637 . . . 4 Rel ∅
3 wesn 5605 . . . 4 (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅))
42, 3ax-mp 5 . . 3 (∅ We {∅} ↔ ¬ ∅∅∅)
51, 4mpbir 234 . 2 ∅ We {∅}
6 df1o2 8102 . . 3 1o = {∅}
7 weeq2 5509 . . 3 (1o = {∅} → (∅ We 1o ↔ ∅ We {∅}))
86, 7ax-mp 5 . 2 (∅ We 1o ↔ ∅ We {∅})
95, 8mpbir 234 1 ∅ We 1o
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538  ∅c0 4243  {csn 4525   class class class wbr 5031   We wwe 5478  Rel wrel 5525  1oc1o 8081 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-suc 6166  df-1o 8088 This theorem is referenced by:  psr1tos  20828
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