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Theorem dford2 9612
Description: Assuming ax-reg 9584, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.)
Assertion
Ref Expression
dford2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dford2
StepHypRef Expression
1 df-ord 6365 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2 zfregfr 9597 . . . . 5 E Fr 𝐴
3 dfwe2 7758 . . . . 5 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
42, 3mpbiran 708 . . . 4 ( E We 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
5 epel 5583 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
6 biid 261 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
7 epel 5583 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
85, 6, 73orbi123i 1157 . . . . 5 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
982ralbii 3129 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
104, 9bitri 275 . . 3 ( E We 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
1110anbi2i 624 . 2 ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
121, 11bitri 275 1 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3o 1087  wral 3062   class class class wbr 5148  Tr wtr 5265   E cep 5579   Fr wfr 5628   We wwe 5630  Ord word 6361
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-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722  ax-reg 9584
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-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6365
This theorem is referenced by:  ordelordALT  43284  ordelordALTVD  43614
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