Theorem ordon 4212

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 variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tron 4119	. 2 ⊢ Tr On
2		df-on 4105	. . . . 5 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	2	abeq2i 2148	. . . 4 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
4		ordtr 4115	. . . 4 ⊢ (Ord 𝑥 → Tr 𝑥)
5	3, 4	sylbi 114	. . 3 ⊢ (𝑥 ∈ On → Tr 𝑥)
6	5	rgen 2374	. 2 ⊢ ∀𝑥 ∈ On Tr 𝑥
7		dford3 4104	. 2 ⊢ (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥))
8	1, 6, 7	mpbir2an 849	1 ⊢ Ord On

Syntax hints:   ∈ wcel 1393  ∀wral 2306  Tr wtr 3854  Ord word 4099  Oncon0 4100

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105

This theorem is referenced by:  ssorduni  4213  limon  4239  onprc  4276