Theorem tron 4403

Description: The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)

Assertion

Ref	Expression
tron	⊢ Tr On

Proof of Theorem tron

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr3 4123	. 2 ⊢ (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On)
2		vex 2755	. . . . . . 7 ⊢ 𝑥 ∈ V
3	2	elon 4395	. . . . . 6 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
4		ordelord 4402	. . . . . 6 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
5	3, 4	sylanb 284	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
6	5	ex 115	. . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → Ord 𝑦))
7		vex 2755	. . . . 5 ⊢ 𝑦 ∈ V
8	7	elon 4395	. . . 4 ⊢ (𝑦 ∈ On ↔ Ord 𝑦)
9	6, 8	imbitrrdi 162	. . 3 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On))
10	9	ssrdv 3176	. 2 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
11	1, 10	mprgbir 2548	1 ⊢ Tr On

Syntax hints:   ∈ wcel 2160   ⊆ wss 3144  Tr wtr 4119  Ord word 4383  Oncon0 4384

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 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-uni 3828  df-tr 4120  df-iord 4387  df-on 4389

This theorem is referenced by:  ordon  4506  tfi  4602