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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 6375 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | ordelss 6381 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴) | |
3 | 2 | ex 414 | . 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 2107 ⊆ wss 3949 Ord word 6364 Oncon0 6365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-v 3477 df-in 3956 df-ss 3966 df-uni 4910 df-tr 5267 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-ord 6368 df-on 6369 |
This theorem is referenced by: ordunidif 6414 onelssi 6480 ssorduni 7766 sucexeloniOLD 7798 suceloniOLD 7800 tfisi 7848 poseq 8144 tfrlem9 8385 tfrlem11 8388 oaordex 8558 oaass 8561 odi 8579 omass 8580 oewordri 8592 nnaordex 8638 domtriord 9123 hartogs 9539 card2on 9549 tskwe 9945 infxpenlem 10008 cfub 10244 cfsuc 10252 coflim 10256 hsmexlem2 10422 ondomon 10558 pwcfsdom 10578 inar1 10770 tskord 10775 grudomon 10812 gruina 10813 sltres 27165 nosupno 27206 nosupbday 27208 noinfno 27221 oldssmade 27372 madebday 27394 mulsproplem13 27584 mulsproplem14 27585 dfrdg2 34767 aomclem6 41801 nnoeomeqom 42062 naddgeoa 42145 naddwordnexlem1 42148 naddwordnexlem4 42152 iscard5 42287 |
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