Theorem ordon 4397

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

Step	Hyp	Ref	Expression
1		tron 4299	. 2 ⊢ Tr On
2		df-on 4285	. . . . 5 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	2	abeq2i 2248	. . . 4 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
4		ordtr 4295	. . . 4 ⊢ (Ord 𝑥 → Tr 𝑥)
5	3, 4	sylbi 120	. . 3 ⊢ (𝑥 ∈ On → Tr 𝑥)
6	5	rgen 2483	. 2 ⊢ ∀𝑥 ∈ On Tr 𝑥
7		dford3 4284	. 2 ⊢ (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥))
8	1, 6, 7	mpbir2an 926	1 ⊢ Ord On

Syntax hints:   ∈ wcel 1480  ∀wral 2414  Tr wtr 4021  Ord word 4279  Oncon0 4280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285

This theorem is referenced by:  ssorduni  4398  limon  4424  onprc  4462  tfri1dALT  6241  rdgon  6276