Theorem dford2 9611

Description: Assuming ax-reg 9583, 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 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dford2

Step	Hyp	Ref	Expression
1		df-ord 6357	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
2		zfregfr 9596	. . . . 5 ⊢ E Fr 𝐴
3		dfwe2 7754	. . . . 5 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))
4	2, 3	mpbiran 706	. . . 4 ⊢ ( E We 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
5		epel 5573	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
6		biid 261	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
7		epel 5573	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
8	5, 6, 7	3orbi123i 1153	. . . . 5 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
9	8	2ralbii 3120	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
10	4, 9	bitri 275	. . 3 ⊢ ( E We 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
11	10	anbi2i 622	. 2 ⊢ ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
12	1, 11	bitri 275	1 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∨ w3o 1083  ∀wral 3053   class class class wbr 5138  Tr wtr 5255   E cep 5569   Fr wfr 5618   We wwe 5620  Ord word 6353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718  ax-reg 9583

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6357

This theorem is referenced by:  ordelordALT  43787  ordelordALTVD  44117