Theorem ordon 4518

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 4413	. 2 ⊢ Tr On
2		df-on 4399	. . . . 5 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	2	abeq2i 2304	. . . 4 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
4		ordtr 4409	. . . 4 ⊢ (Ord 𝑥 → Tr 𝑥)
5	3, 4	sylbi 121	. . 3 ⊢ (𝑥 ∈ On → Tr 𝑥)
6	5	rgen 2547	. 2 ⊢ ∀𝑥 ∈ On Tr 𝑥
7		dford3 4398	. 2 ⊢ (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥))
8	1, 6, 7	mpbir2an 944	1 ⊢ Ord On

Syntax hints:   ∈ wcel 2164  ∀wral 2472  Tr wtr 4127  Ord word 4393  Oncon0 4394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3159  df-ss 3166  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399

This theorem is referenced by:  ssorduni  4519  limon  4545  onprc  4584  tfri1dALT  6404  rdgon  6439