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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 6216 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | |
3 | 1, 2 | mpan 688 | 1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Oncon0 6191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 |
This theorem is referenced by: onssneli 6300 oawordeulem 8180 rankuni 9292 tcrank 9313 cardne 9394 cardval2 9420 alephsuc2 9506 cfsmolem 9692 cfcof 9696 alephreg 10004 pwcfsdom 10005 tskcard 10203 onsucconni 33785 |
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