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Theorem 0we1 8505
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 5197 . . 3 ¬ ∅∅∅
2 rel0 5799 . . . 4 Rel ∅
3 wesn 5764 . . . 4 (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅))
42, 3ax-mp 5 . . 3 (∅ We {∅} ↔ ¬ ∅∅∅)
51, 4mpbir 230 . 2 ∅ We {∅}
6 df1o2 8472 . . 3 1o = {∅}
7 weeq2 5665 . . 3 (1o = {∅} → (∅ We 1o ↔ ∅ We {∅}))
86, 7ax-mp 5 . 2 (∅ We 1o ↔ ∅ We {∅})
95, 8mpbir 230 1 ∅ We 1o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1541  c0 4322  {csn 4628   class class class wbr 5148   We wwe 5630  Rel wrel 5681  1oc1o 8458
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-suc 6370  df-1o 8465
This theorem is referenced by:  psr1tos  21712
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