Theorem tron 6385

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 5224	. 2 ⊢ (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On)
2		vex 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
3	2	elon 6371	. . . . . 6 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
4		ordelord 6384	. . . . . 6 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
5	3, 4	sylanb 592	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
6	5	ex 417	. . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → Ord 𝑦))
7		vex 3459	. . . . 5 ⊢ 𝑦 ∈ V
8	7	elon 6371	. . . 4 ⊢ (𝑦 ∈ On ↔ Ord 𝑦)
9	6, 8	imbitrrdi 255	. . 3 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On))
10	9	ssrdv 3944	. 2 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
11	1, 10	mprgbir 3086	1 ⊢ Tr On

Syntax hints:   ∈ wcel 2143   ⊆ wss 3906  Tr wtr 5219  Ord word 6361  Oncon0 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6365  df-on 6366

This theorem is referenced by:  ordon  7777  predon  7786  onuninsuci  7837  gruina  10804