Theorem tron 6365

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 5211	. 2 ⊢ (Tr On ↔ ∀𝑥 ∈ On 𝑥 ⊆ On)
2		vex 3457	. . . . . . 7 ⊢ 𝑥 ∈ V
3	2	elon 6351	. . . . . 6 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
4		ordelord 6364	. . . . . 6 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
5	3, 4	sylanb 590	. . . . 5 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
6	5	ex 416	. . . 4 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → Ord 𝑦))
7		vex 3457	. . . . 5 ⊢ 𝑦 ∈ V
8	7	elon 6351	. . . 4 ⊢ (𝑦 ∈ On ↔ Ord 𝑦)
9	6, 8	imbitrrdi 254	. . 3 ⊢ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ On))
10	9	ssrdv 3942	. 2 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
11	1, 10	mprgbir 3082	1 ⊢ Tr On

Syntax hints:   ∈ wcel 2141   ⊆ wss 3904  Tr wtr 5206  Ord word 6341  Oncon0 6342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-tr 5207  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-ord 6345  df-on 6346

This theorem is referenced by:  ordon  7756  predon  7765  onuninsuci  7816  gruina  10773