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Theorem onelord 43198
Description: Every element of a ordinal is an ordinal. Lemma 1.3 of [Schloeder] p. 1. Based on onelon 6405 and eloni 6390. (Contributed by RP, 15-Jan-2025.)
Assertion
Ref Expression
onelord ((𝐴 ∈ On ∧ 𝐵𝐴) → Ord 𝐵)

Proof of Theorem onelord
StepHypRef Expression
1 onelon 6405 . 2 ((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ On)
2 eloni 6390 . 2 (𝐵 ∈ On → Ord 𝐵)
31, 2syl 17 1 ((𝐴 ∈ On ∧ 𝐵𝐴) → Ord 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2104  Ord word 6379  Oncon0 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ne 2937  df-ral 3058  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-ord 6383  df-on 6384
This theorem is referenced by: (None)
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