Theorem ordon 4362

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 4264	. 2 ⊢ Tr On
2		df-on 4250	. . . . 5 ⊢ On = {𝑥 ∣ Ord 𝑥}
3	2	abeq2i 2225	. . . 4 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
4		ordtr 4260	. . . 4 ⊢ (Ord 𝑥 → Tr 𝑥)
5	3, 4	sylbi 120	. . 3 ⊢ (𝑥 ∈ On → Tr 𝑥)
6	5	rgen 2459	. 2 ⊢ ∀𝑥 ∈ On Tr 𝑥
7		dford3 4249	. 2 ⊢ (Ord On ↔ (Tr On ∧ ∀𝑥 ∈ On Tr 𝑥))
8	1, 6, 7	mpbir2an 909	1 ⊢ Ord On

Syntax hints:   ∈ wcel 1463  ∀wral 2390  Tr wtr 3986  Ord word 4244  Oncon0 4245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-in 3043  df-ss 3050  df-uni 3703  df-tr 3987  df-iord 4248  df-on 4250

This theorem is referenced by:  ssorduni  4363  limon  4389  onprc  4427  tfri1dALT  6202  rdgon  6237