Theorem tron 6325

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 5201	. 2 ⊢ (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On)
2		vex 3438	. . . . . . 7 ⊢ 𝑥 ∈ V
3	2	elon 6311	. . . . . 6 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
4		ordelord 6324	. . . . . 6 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
5	3, 4	sylanb 581	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
6	5	ex 412	. . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → Ord 𝑦))
7		vex 3438	. . . . 5 ⊢ 𝑦 ∈ V
8	7	elon 6311	. . . 4 ⊢ (𝑦 ∈ On ↔ Ord 𝑦)
9	6, 8	imbitrrdi 252	. . 3 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On))
10	9	ssrdv 3938	. 2 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
11	1, 10	mprgbir 3052	1 ⊢ Tr On

Syntax hints:   ∈ wcel 2110   ⊆ wss 3900  Tr wtr 5196  Ord word 6301  Oncon0 6302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-tr 5197  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6305  df-on 6306

This theorem is referenced by:  ordon  7705  predon  7714  onuninsuci  7765  gruina  10701