Theorem ordon 4470

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 4367	. 2 ⊢ Tr On
2		df-on 4353	. . . . 5 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	2	abeq2i 2281	. . . 4 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
4		ordtr 4363	. . . 4 ⊢ (Ord 𝑥 → Tr 𝑥)
5	3, 4	sylbi 120	. . 3 ⊢ (𝑥 ∈ On → Tr 𝑥)
6	5	rgen 2523	. 2 ⊢ ∀𝑥 ∈ On Tr 𝑥
7		dford3 4352	. 2 ⊢ (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥))
8	1, 6, 7	mpbir2an 937	1 ⊢ Ord On

Syntax hints:   ∈ wcel 2141  ∀wral 2448  Tr wtr 4087  Ord word 4347  Oncon0 4348

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by:  ssorduni  4471  limon  4497  onprc  4536  tfri1dALT  6330  rdgon  6365