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Theorem 0we1 7755
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 4853 . . 3 ¬ ∅∅∅
2 rel0 5399 . . . 4 Rel ∅
3 wesn 5347 . . . 4 (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅))
42, 3ax-mp 5 . . 3 (∅ We {∅} ↔ ¬ ∅∅∅)
51, 4mpbir 221 . 2 ∅ We {∅}
6 df1o2 7741 . . 3 1𝑜 = {∅}
7 weeq2 5255 . . 3 (1𝑜 = {∅} → (∅ We 1𝑜 ↔ ∅ We {∅}))
86, 7ax-mp 5 . 2 (∅ We 1𝑜 ↔ ∅ We {∅})
95, 8mpbir 221 1 ∅ We 1𝑜
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1632  c0 4058  {csn 4321   class class class wbr 4804   We wwe 5224  Rel wrel 5271  1𝑜c1o 7722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-suc 5890  df-1o 7729
This theorem is referenced by:  psr1tos  19761
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