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Theorem dford2 9071
Description: Assuming ax-reg 9044, 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 6187 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2 zfregfr 9056 . . . . 5 E Fr 𝐴
3 dfwe2 7485 . . . . 5 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
42, 3mpbiran 705 . . . 4 ( E We 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
5 epel 5462 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
6 biid 262 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
7 epel 5462 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
85, 6, 73orbi123i 1148 . . . . 5 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
982ralbii 3163 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
104, 9bitri 276 . . 3 ( E We 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
1110anbi2i 622 . 2 ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
121, 11bitri 276 1 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3o 1078  wral 3135   class class class wbr 5057  Tr wtr 5163   E cep 5457   Fr wfr 5504   We wwe 5506  Ord word 6183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450  ax-reg 9044
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187
This theorem is referenced by:  ordelordALT  40748  ordelordALTVD  41078
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