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| Mirrors > Home > MPE Home > Th. List > oneli | Structured version Visualization version GIF version | ||
| Description: A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) | 
| Ref | Expression | 
|---|---|
| on.1 | ⊢ 𝐴 ∈ On | 
| Ref | Expression | 
|---|---|
| oneli | ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | on.1 | . 2 ⊢ 𝐴 ∈ On | |
| 2 | onelon 6409 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | |
| 3 | 1, 2 | mpan 690 | 1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 Oncon0 6384 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 | 
| This theorem is referenced by: onssneli 6500 oawordeulem 8592 rankuni 9903 tcrank 9924 cardne 10005 cardval2 10031 alephsuc2 10120 cfsmolem 10310 cfcof 10314 alephreg 10622 pwcfsdom 10623 tskcard 10821 lrcut 27941 onsucconni 36438 | 
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