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Theorem dford2 8468
Description: Assuming ax-reg 8448, 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 5690 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2 zfregfr 8460 . . . . 5 E Fr 𝐴
3 dfwe2 6935 . . . . 5 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
42, 3mpbiran 952 . . . 4 ( E We 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
5 epel 4993 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
6 biid 251 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
7 epel 4993 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
85, 6, 73orbi123i 1250 . . . . 5 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
982ralbii 2976 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
104, 9bitri 264 . . 3 ( E We 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
1110anbi2i 729 . 2 ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
121, 11bitri 264 1 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3o 1035  wral 2907   class class class wbr 4618  Tr wtr 4717   E cep 4988   Fr wfr 5035   We wwe 5037  Ord word 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909  ax-reg 8448
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690
This theorem is referenced by:  ordelordALT  38256  ordelordALTVD  38613
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