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| Mirrors > Home > MPE Home > Th. List > onelss | Structured version Visualization version GIF version | ||
| Description: An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
| Ref | Expression |
|---|---|
| onelss | ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eloni 6394 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 2 | ordelss 6400 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴) | |
| 3 | 2 | ex 412 | . 2 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
| 4 | 1, 3 | syl 17 | 1 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ⊆ wss 3951 Ord word 6383 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-ss 3968 df-uni 4908 df-tr 5260 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: ordunidif 6433 onelssi 6499 ssorduni 7799 sucexeloniOLD 7830 suceloniOLD 7832 tfisi 7880 poseq 8183 tfrlem9 8425 tfrlem11 8428 oaordex 8596 oaass 8599 odi 8617 omass 8618 oewordri 8630 nnaordex 8676 domtriord 9163 hartogs 9584 card2on 9594 tskwe 9990 infxpenlem 10053 cfub 10289 cfsuc 10297 coflim 10301 hsmexlem2 10467 ondomon 10603 pwcfsdom 10623 inar1 10815 tskord 10820 grudomon 10857 gruina 10858 sltres 27707 nosupno 27748 nosupbday 27750 noinfno 27763 oldssmade 27916 madebday 27938 mulsproplem13 28154 mulsproplem14 28155 dfrdg2 35796 aomclem6 43071 nnoeomeqom 43325 naddgeoa 43407 naddwordnexlem1 43410 naddwordnexlem4 43414 iscard5 43549 |
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