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Mirrors > Home > MPE Home > Th. List > Mathboxes > onelord | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
onelord | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → Ord 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onelon 6405 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | |
2 | eloni 6390 | . 2 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
3 | 1, 2 | syl 17 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 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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