Theorem tron 4312

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 4038	. 2 ⊢ (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On)
2		vex 2692	. . . . . . 7 ⊢ 𝑥 ∈ V
3	2	elon 4304	. . . . . 6 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
4		ordelord 4311	. . . . . 6 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
5	3, 4	sylanb 282	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
6	5	ex 114	. . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → Ord 𝑦))
7		vex 2692	. . . . 5 ⊢ 𝑦 ∈ V
8	7	elon 4304	. . . 4 ⊢ (𝑦 ∈ On ↔ Ord 𝑦)
9	6, 8	syl6ibr 161	. . 3 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On))
10	9	ssrdv 3108	. 2 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
11	1, 10	mprgbir 2493	1 ⊢ Tr On

Syntax hints:   ∈ wcel 1481   ⊆ wss 3076  Tr wtr 4034  Ord word 4292  Oncon0 4293

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-uni 3745  df-tr 4035  df-iord 4296  df-on 4298

This theorem is referenced by:  ordon  4410  tfi  4504