Theorem ordon 7024

Description: The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)

Assertion

Ref	Expression
ordon	⊢ Ord On

Proof of Theorem ordon

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tron 5784	. 2 ⊢ Tr On
2		onfr 5801	. . 3 ⊢ E Fr On
3		eloni 5771	. . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥)
4		eloni 5771	. . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦)
5		ordtri3or 5793	. . . . . 6 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
6		epel 5061	. . . . . . 7 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
7		biid 251	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
8		epel 5061	. . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
9	6, 7, 8	3orbi123i 1271	. . . . . 6 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
10	5, 9	sylibr 224	. . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
11	3, 4, 10	syl2an 493	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
12	11	rgen2a 3006	. . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)
13		dfwe2 7023	. . 3 ⊢ ( E We On ↔ ( E Fr On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))
14	2, 12, 13	mpbir2an 975	. 2 ⊢ E We On
15		df-ord 5764	. 2 ⊢ (Ord On ↔ (Tr On ∧ E We On))
16	1, 14, 15	mpbir2an 975	1 ⊢ Ord On

Syntax hints:   ∧ wa 383   ∨ w3o 1053   ∈ wcel 2030  ∀wral 2941   class class class wbr 4685  Tr wtr 4785   E cep 5057   Fr wfr 5099   We wwe 5101  Ord word 5760  Oncon0 5761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765

This theorem is referenced by:  epweon  7025  onprc  7026  ssorduni  7027  ordeleqon  7030  ordsson  7031  onint  7037  suceloni  7055  limon  7078  tfi  7095  ordom  7116  ordtypelem2  8465  hartogs  8490  card2on  8500  tskwe  8814  alephsmo  8963  ondomon  9423  dford3lem2  37911  dford3  37912  iunord  42747