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Theorem 0we1 8456
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 5158 . . 3 ¬ ∅∅∅
2 rel0 5759 . . . 4 Rel ∅
3 wesn 5724 . . . 4 (Rel ∅ → (∅ We {∅} ↔ ¬ ∅∅∅))
42, 3ax-mp 5 . . 3 (∅ We {∅} ↔ ¬ ∅∅∅)
51, 4mpbir 230 . 2 ∅ We {∅}
6 df1o2 8423 . . 3 1o = {∅}
7 weeq2 5626 . . 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 1542  c0 4286  {csn 4590   class class class wbr 5109   We wwe 5591  Rel wrel 5642  1oc1o 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-suc 6327  df-1o 8416
This theorem is referenced by:  psr1tos  21583
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