Theorem ordon 7150

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 5900	. 2 ⊢ Tr On
2		onfr 5917	. . 3 ⊢ E Fr On
3		eloni 5887	. . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥)
4		eloni 5887	. . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦)
5		ordtri3or 5909	. . . . . 6 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
6		epel 5179	. . . . . . 7 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
7		biid 252	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
8		epel 5179	. . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
9	6, 7, 8	3orbi123i 1186	. . . . . 6 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
10	5, 9	sylibr 225	. . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
11	3, 4, 10	syl2an 584	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
12	11	rgen2a 3129	. . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)
13		dfwe2 7149	. . 3 ⊢ ( E We On ↔ ( E Fr On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))
14	2, 12, 13	mpbir2an 691	. 2 ⊢ E We On
15		df-ord 5880	. 2 ⊢ (Ord On ↔ (Tr On ∧ E We On))
16	1, 14, 15	mpbir2an 691	1 ⊢ Ord On

Syntax hints:   ∧ wa 383   ∨ w3o 1097   ∈ wcel 2148  ∀wral 3064   class class class wbr 4797  Tr wtr 4899   E cep 5175   Fr wfr 5219   We wwe 5221  Ord word 5876  Oncon0 5877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pr 5048  ax-un 7117

This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-tr 4900  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-ord 5880  df-on 5881

This theorem is referenced by:  epweon  7151  onprc  7152  ssorduni  7153  ordeleqon  7156  ordsson  7157  onint  7163  suceloni  7181  limon  7204  tfi  7221  ordom  7242  ordtypelem2  8601  hartogs  8626  card2on  8636  tskwe  8997  alephsmo  9146  ondomon  9608  dford3lem2  38135  dford3  38136  iunord  42974