Theorem ontr1 4479

Description: Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)

Assertion

Ref	Expression
ontr1	⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))

Proof of Theorem ontr1

Step	Hyp	Ref	Expression
1		eloni 4465	. 2 ⊢ (𝐶 ∈ On → Ord 𝐶)
2		ordtr1 4478	. 2 ⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
3	1, 2	syl 14	1 ⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2200  Ord word 4452  Oncon0 4453

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-uni 3888  df-tr 4182  df-iord 4456  df-on 4458

This theorem is referenced by:  smoiun  6445  nntr2  6647  onunsnss  7075  snon0  7098  exmidontriimlem2  7400  ltsopi  7503  prarloclemarch2  7602  pwle2  16323